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Excerpts on the Euler-Maclaurin summation formula, fromInstitutiones Calculi Differentialis
Excerpts on the Euler-Maclaurin summation formula, fromInstitutiones Calculi Differentialis by Leonhard Euler.
Translated by David PengelleyCopyright c 2000 by David Pengelley
(individual educational use only may be made without permission)
Leonhard Eulers book Institutiones Calculi Differentialis (Foundationsof Differential Calculus) was published in two parts in 1755 [2, series I, vol.10], and translated into German in 1790 [3]. There is an English translationof part 1 [4], but not part 2, of the Institutiones.Euler was entranced by infinite series, and a wizard at working with them.
A lot of the book is actually devoted to the relationship between differentialcalculus and infinite series, and in this respect it differs considerably fromtodays calculus books. In chapters 5 and 6 of part 2 Euler presents his wayof finding sums of series, both finite and infinite, via his discovery of theEuler-Maclaurin summation formula.Here we present translated excerpts from these two chapters of part 2,
featuring aspects of Eulers development and applications of his summationformula. The English translation has been made primarily from the Germantranslation of 1790 [3], with some assistance from Daniel Otero in comparingthe resulting English with the Latin original. The excerpts include those ina forthcoming book of annotated original sources, within a chapter on TheBridge Between the Continuous and Discrete. The book chapter followsthis theme via sources by Archimedes, Fermat, Pascal, Bernoulli and Euler.For more information see http://math.nmsu.edu/~history. In the spiritof providing pure uninterpreted translation, we have here removed all ourannotation and commentary, which along with extensive exercises can befound in the book chapter. The version with annotation and exercises maybe provided at this site at a later time. The only commentary that remainshere summarizes some of those portions of Chapters 5 and 6 that were nottranslated.In excerpts from chapter 5 we see Euler derive his summation formula,
analyze the nature of its Bernoulli numbers in connection with trigonometricfunctions, find the precise sums of infinite series of reciprocal even powers,and prove Bernoullis sums of powers formulas. From chapter 6 we see threediverse applications of the summation formula, each revealing a fundamen-tally different way of using it. We first see Euler approximate large partialsums of the slowly diverging harmonic series
i=1
1i , which involves approx-
imating the now famous Euler constant. Then we see how in the early1730s Euler approximated the infinite sum of reciprocal squares to greatprecision without knowledge of the infinite sum itself. Finally Euler goes onto use the summation formula to study sums of logarithms, from which heobtains incredibly impressive formulas and approximations for large facto-rials (Stirlings series), and thence for binomial coefficients, using Wallissformula for pi to determine the unknown constant in his summation formula.
1
2Leonhard Euler, fromFoundations of Differential Calculus
Part Two, Chapter 5On Finding Sums of Series from the General Term
103. Suppose y is the general term of a series, belonging to the index x,and thus y is any function of x. Further, suppose Sy is the summative term ofthis series, expressing the aggregate of all terms from the first or another fixedterm up to y, inclusive. The sums of the series are calculated from the firstterm, so that if x = 1, y is the first term, and likewise Sy yields this first term;alternatively, if x = 0, the summative term Sy vanishes, because no terms arebeing summed. With these stipulations, the summative term Sy is a functionof x that vanishes if one sets x = 0.[...]
105. Consider a series whose general term, belonging to the index x, is y,and whose preceding term, with index x1, is v; because v arises from y, whenx is replaced by x 1, one has
v = y dydx
+ddy
2dx2 d
3y
6dx3+
d4y
24dx4 d
5y
120dx5+ etc.
If y is the general term of the series
1 2 3 4 x 1 xa+ b+ c+ d+ + v + y
and if the term belonging to the index 0 is A, then v, as a function of x, is thegeneral term of the series
1 2 3 4 5 xA+ a+ b+ c+ d+ + v ,
so if Sv denotes the sum of this series, then Sv = Sy y + A. If one setsx = 0, then Sy = 0 and y = A, so Sv vanishes.106. Because
v = y dydx
+ddy
2dx2 d
3y
6dx3+ etc.,
one has, from the preceding,
Sv = Sy S dydx
+ Sddy
2dx2 S d
3y
6dx3+ S
d4y
24dx4 etc.,
and, because Sv = Sy y +A,
y A = S dydx
S ddy2dx2
+ Sd3y
6dx3 S d
4y
24dx4+ etc.,
or equivalently
Sdy
dx= y A+ S ddy
2dx2 S d
3y
6dx3+ S
d4y
24dx4 etc.
3Thus if one knows the sums of the series, whose general terms are ddydx2
, d3ydx3
, d4ydx4
,
etc., one can obtain the summative term of the series whose general term is dydx .
The constant A must then be such that the summative term S dydx disappearswhen x = 0, and this condition makes it easier to determine, than saying thatit is the term belonging to the index 0 in the series whose general term is y.
In 107/108 Euler illustrates the practical application of this equation bychoosing to use the power function y = xn+1/(n+1). This has the advantagethat the derivatives in the equation are just lower power functions, so thatthe sums are all sums of powers, and then vanish after some point in theequation, so he obtains a finite expression for Sxn (i.e., for
xi=1 i
n). Heapplies this inductively from n = 0 upwards to calculate the closed formulaefor sums of powers of the natural numbers explicitly up through the sum offourth powers.[...]
109. Since from the above one has
Sdy
dx= y [A] + 1
2Sddy
dx2 16Sd3y
dx3+
124Sd4y
dx4 1120
Sd5y
dx5+ etc.,
if one sets dydx = z, thenddydx2
= dzdx ,d3ydx3
= ddzdx2
, etc. And because dy = zdx, y willbe a quantity whose differential is zdx, and this one writes as y =
zdx. Now
the determination of the quantity y from z according to this formula assumesthe integral calculus; but we can nevertheless make use of this expression
zdx,
if for z we use no function other than that whose differential is zdx from above.Thus substituting these values yields
Sz =zdx+
12Sdz
dx 16Sddz
dx2+
124Sd3z
dx3 etc.,
adding to it a constant value such that when x = 0, the sum Sz also vanishes.110. But if in the expressions above one substitutes the letter z in place of
y, or if one differentiates the preceding equation, which yields the same, oneobtains
Sdz
dx= z +
12Sddz
dx2 16Sd3z
dx3+
124Sd4z
dx4 etc.;
but using dzdx in place of y one obtains
Sddz
dx2=
dz
dx+12Sd3z
dx3 16Sd4z
dx4+
124Sd5z
dx5 etc.
Similarly replacing y successively by the values ddzdx2
, d3zdx3
etc., produces
Sd3z
dx3=ddz
dx2+12Sd4z
dx4 16Sd5z
dx5+
124Sd6z
dx6 etc.,
Sd4z
dx4=
d3z
dx3+12Sd5z
dx5 16Sd6z
dx6+
124Sd7z
dx7 etc.,
and so forth indefinitely.
4111. Now when these values for S dzdx , Sddzdx2
, S d3zdx3
are successively substitutedin the expression
Sz =zdx+
12Sdz
dx 16Sddz
dx2+
124Sd3z
dx3 etc.,
one finds an expression for Sz, composed of the termszdx, z, dzdx ,
ddzdx2
, d3zdx3
etc., whose coefficients are easily obtained as follows. One sets
Sz =zdx+ z +
dz
dx+ddz
dx2+d3z
dx3+d4z
dx4+ etc.,
and substitutes for these terms the values they have from the previous series,yielding
zdx=Sz 12S dzdx + 16S ddzdx2 124S d3zdx3
+ 1120Sd4zdx4
etc.z= +S dzdx 2S ddzdx2 + 6S d
3zdx3
24S d4zdz4
+ etc.dzdx = S
ddzdx2
2S d3zdx3
+ 6Sd4zdx4
etc.ddzdx2
= S d3zdx3
2S d4zdx4
+ etc.d3zdx3
= S d4zdx4
etc.etc.
Since these values, added together, must produce Sz, the coefficients , , , etc. are defined by the sequence of equations
12= 0,
2+16= 0,
2+
6 124
= 0,
2+
6 24
+1120
= 0, 2+
6 24
+
120 1720
= 0,
2+
6 24
+
120 720
+1
5040= 0 etc.
112. So from these equations the successive values of all the letters , , , etc. are defined; they are
=12, =
2 16=
112, =
2
6+
124
= 0,
=
2
6+
24 1120
= 1720
, =
2 6+
24 120
+1720
= 0 etc.,
and if one continues in this fashion one finds that alternating terms vanish. Thethird, fifth, seventh letters, and so on, in fact all odd terms except the first, arezero, so that this series appears to contradict the law of continuity by which theterms proceed. A rigorous proof is especially needed that all odd terms exceptthe first vanish.
5113. Because the letters are determined from the preceding by a constantlaw, they form a recurrent series. In order to develop this, consider the series
1 + u+ u2 + u3 + u4 + u5 + u6 + etc.,
and set its value = V , so it is clear that this recurrent series arises from thedevelopment of the fraction
V =1
1 12u+ 16u2 124u3 + 1120u4 etc.And when this fraction is resolved in a different way in an infinite series accordingto the powers of u, then necessarily the same series
V = 1 + u+ u2 + u3 + u4 + u5 + etc.
will always result. In this fashion a different rule for determining the letters ,, , etc. results.
114. Because one has
eu = 1 u+ 12u2 1
6u3 +
124u4 1
120u5 + etc.,
where e denotes the number whose hyperbolic logarithm is one, then
1 euu
= 1 12u+
16u2 1
24u3 +
1120
u4 etc.,and thus
V =u
1 eu .Now one removes from this series the second term u = 12u, so that
V 12u = 1 + u2 + u3 + u4 + u5 + u6 + etc.;
whence
V 12u =
12u (1 + e
u)1 eu .
Multiplying numerator and denominator by e12u yields
V 12u =
u(e12u + e
12u)
2(e12u e 12u
) ,and converting the quantities e
12u and e
12u into series gives
V 12u =
1 + u2
24 +u4
2468 +u6
24681012 + etc.
2(12 +
u2
246 +u4
246810 + etc.)
or
V 12u =
1 + u2
24 +u4
2468 +u6
2412 +u8
2416 + etc.1 + u246 +
u4
46810 +u6
4614 +u8
4618 + etc.
6115. Since no odd powers occur in this fraction, likewise none can occur inits expansion; because V 12u equals the series
1 + u2 + u3 + u4 + u5 + u6 + etc.,
the coefficients of the odd powers , , , etc. all vanish. And so it isclear why the even-ordered terms after the second all equal zero in the series1 + u+ u2 + u3 + u4+ etc., for otherwise the law of continuity would beviolated. Thus
V = 1 +12u+ u2 + u4 + u6 + u8 + u10 + etc.,
and if the letters , , , , have been determined by the development of theabove fraction, one obtains the summative term Sz of the series, whose generalterm = z corresponds to the index x, expressed as
Sz =zdx+
12z +
dz
dx+d3z
dx3+d5z
dx5+d7z
dx7+ etc.
116. Since the series 1 + u2 + u4 + u6 + u8+ etc. arises by developingthe fraction
1 + u2
24 +u4
2468 +u6
24681012 + etc.1 + u246 +
u4
46810 +u6
468101214 + etc.,
the letters , , , , x will follow according to the rule, as
= 124 146 = 12468 46 146810 = 124612 46 46810 14614 = 124616 46 46810 4614 14618
etc.
But these values are alternatingly positive and negative.
117. If the letters are alternatingly taken negatively, so that
Sz =zdx+
12z dz
dx+d3z
dx3 d
5z
dx5+d7z
dx7 etc.,
then the letters , , , , etc. are determined by the fraction
1 u224 + u4
2468 u6
2412 +u8
2416 etc.1 u246 + u
4
46810 u6
4614 +u8
4618 etc.,
when one develops it to the series
1 + u2 + u4 + u6 + u8 + etc.
7From this one has
=1
4 6 1
2 4 =
4 6 1
4 6 8 10 +1
2 4 6 8 =
4 6
4 6 8 10 +1
4 6 14 1
2 4 12etc.;
only here all terms are negative.
118. Thus we consider = A; = B; = C, etc.; consequently
Sz =zdx+
12z +
Adz
dx Bd
3z
dx3+Cd5z
dx5 Dd
7z
dx7+ etc.,
and in order to determine the letters A, B, C, D etc., we consider the series
1Au2 Bu4 Cu6 Du8 Eu10 etc.,which arises from the development of the fraction
1 u224 + u4
2468 u6
2412 +u8
2416 etc.1 u246 + u
4
46810 u6
4614 +u8
4618 etc.,
or consider the series1uAuBu3 Cu5 Du7 Eu9 etc. = s,
which arises from the development of the fraction
s =1 u224 + u
4
2468 u6
2412 + etc.u u346 + u
5
46810 u7
4614 + etc..
But since
cos12u = 1 u
2
2 4 +u4
2 4 6 8 u6
2 4 12 + etc.,
sin12u =
u
2 u
3
2 4 6 +u5
2 4 6 8 10 u7
2 4 14 + etc.,we have
s =cos 12u2 sin 12u
=12cot
12u.
Thus if one converts the cotangent of the arc 12u into a series, according to thepowers of u, the values of the letters A, B, C, D, E, etc. are revealed.119. Because s = 12 cot
12u, one has
12u = A. cot 2s, and if one differentiates,
then 12du =2ds1+4ss , or 4ds+ du+ 4ssdu = 0, or
4dsdu
+ 1 + 4ss = 0.
8But since
s =1uAuBu3 Cu5 etc.,
one has
4dsdu = 4uu 4A 3 4Bu2 5 4Cu4 7 4Du6 etc.1= 1
4ss= 4uu 8A 8Bu2 8Cu4 8Du6 etc.+ 4A2u2+ 8ABu4+ 8ACu6+ etc.
+ 4BBu6+ etc.
Setting the homogeneous terms to zero, one obtains
A =112, B =
A2
5, C =
2AB7
, D =2AC +BB
9, E =
2AD + 2BC11
,
F =2AE + 2BD + CC
13, G =
2AF + 2BE + 2CD15
,
H =2AG+ 2BF + 2CE +DD
17, etc.
From these formulas it is very clearly apparent that each of these values ispositive .
120. But because the denominators of these fractions become very large, andsubstantially impede calculation, we want instead of the letters A, B, C, D,etc. to introduce new ones1:
A =
1 2 3 , B =
1 2 3 4 5 , C =
1 2 3 7 ,
D =
1 2 3 9, E =
1 2 3 11 , etc.
1Translators note: Caution! These new symbols , , ... are completely differentfrom the , , ... used earlier.
9Then one finds
=12, =
232, = 2 3
3, = 2 4
3 +
8 74 5
2,
= 2 53 + 2 10 9 8
1 2 5,
= 2 121 2 3+ 2
12 11 101 2 5 +
12 11 10 9 81 2 7 ,
= 2 141 2 3 + 2
14 13 121 2 5 + 2
14 13 12 11 101 2 7 , etc.
121. But it is more convenient to make use of the formulas
=12, =
43 2, =
63 , = 8
3 + 8 7 6
3 4 5
2,
=103 + 10 9 8
3 4 5 , =123 + 12 11 10
3 4 5 +12 11 10 9 83 4 5 6 7
2,
=143 + 14 13 12
3 4 5 +14 13 12 11 10
3 4 5 6 7 ,
=163 + 16 15 14
3 4 5 +16 15 123 4 7 +
16 15 103 4 9
2
etc.
If one finds the values of the letters , , , , etc. according to this rule,which entails little difficulty in calculation, then one can express the summativeterm of any series, whose general term = z corresponding to the index x, in thefollowing fashion:
Sz =zdx+
12z +
dz
1 2 3 dx d3z
1 2 3 4 5dx3 +d5z
1 2 7dx5
d7z
1 2 9dx7 +d9z
1 2 11dx9 d11z
1 2 13dx11 + etc.
10
As far as the letters , , , , etc. are concerned, one finds the followingvalues:
= 12 or 1 2 = 1 = 16 1 2 3 = 1 = 16 1 2 3 4 = 4 = 310 1 2 3 5 = 36 = 56 1 2 3 6 = 600 = 691210 1 2 3 7 = 24 691 = 352 1 2 3 8 = 20160 35 = 361730 1 2 3 9 = 12096 3617 = 4386742 1 2 3 10 = 86400 43867 = 1222277110 1 2 3 11 = 362880 1222277 = 8545136 1 2 3 12 = 79833600 854513 = 1181820455546 1 2 3 13 = 11404800 1181820455 = 769779272 1 2 3 14 = 43589145600 76977927 = 2374946102930 1 2 3 15 = 43589145600 23749461029pi = 8615841276005462 1 2 3 16pi = 45287424000 8615841276005
etc.
122. These numbers have great use throughout the entire theory of series.First, one can obtain from them the final terms in the sums of even powers,for which we noted above (in 63 of part one) that one cannot obtain them, asone can the other terms, from the sums of earlier powers. For the even powers,the last terms of the sums are products of x and certain numbers, namely forthe 2nd, 4th, 6th, 8th, etc., 16 ,
130 ,
142 ,
130 etc. with alternating signs. But
these numbers arise from the values of the letters , , , , etc., which wefound earlier, when one divides them by the odd numbers 3, 5, 7, 9, etc. Thesenumbers are called the Bernoulli numbers after their discoverer Jakob Bernoulli,
11
and they are3 =
16 =A
19 =
43867798 = I
5 =
130 =B{
21 =174611330 = K =
283617330
7 =
142 =C
23 =
854513138 = L =
111315932323
9 =
130 =D
25 =
2363640912730 =M
11 =
566 =E
27 =
85531036 = N =
136579316
13 =
6912730 =F
29 =
23749461029870 = O
15 =
76 =G
pi31 =
861584127600514322 = P
17 =
3617510 =H etc.
123. Thus one immediately obtains the Bernoulli numbers A, B, C etc. fromthe following equations:
A= 16B= 4312 15 A2C= 6512 27 ABD= 8712 29 AC + 87651234 19B2E= 10912 211 AD + 109871234 211BCF= 121112 213 AE + 12111091234 213BD+ 121110987123456 113 C2G= 141312 215 AF + 141312111234 215BE + 14131211109123456 215 CD
etc.,
and the law of these equations is clear if one notes that whenever the square ofa letter appears, its coefficient is only half as large as it would appear accordingto the rule. But actually one should view the terms containing products ofdifferent letters as occurring twice. So for example
13F =12 111 2 AE+
12 11 10 91 2 3 4 BD+
12 11 10 9 8 71 2 3 4 5 6 CC
+12 11 10 51 2 3 8 DB +
12 11 10 31 2 3 10 EA.
124. Next, the numbers , , , etc. are also ingredients in the expressionsfor the sums of the series of fractions comprised by the general formula
1 +12n
+13n
+14n
+15n
+16n
+ etc.,
when n is a positive even number. We expressed the sums of these series inthe Introductio2 via powers of the semiperiphery pi of the circle of radius = 1,and there one encounters the numbers , , , , etc. in the coefficients of
2Introductio, Book I, chapter 10.
12
these powers. But because these do not appear to occur by accident, rathertheir necessity is apparent, we wish to investigate these sums in a special way,by which the truth of the law of these sums will be clear. Because from above(43) one has
pi
ncot
m
npi =
1m 1nm +
1n+m
12nm +
12n+m
13nm + etc.,
combining terms in pairs one has,
pi
ncot
m
npi =
1m 2mnnm2
2m4n2 m2
2m9n2 m2
2m16n2 m2 etc.,
and from this
1n2 m2 +
14n2 m2 +
19n2 m2 +
116n2 m2 + etc. =
12mm
pi2mn
cotm
npi.
Now we set n = 1 and replace m by u, yielding
11 u2 +
14 u2 +
19 u2 +
116 u2 + etc. =
12uu
pi2u
cotpiu.
Resolving these fractions in series, one obtains
11u2 = 1 + u
2 + u4 + u6 + u8 + etc.
14u2 =
122+ u
2
24+ u
4
26+ u
6
28+ u
8
210+ etc.
19u2 =
132+ u
2
34+ u
4
36+ u
6
38+ u
8
310+ etc.
116u2 =
142+ u
2
44+ u
4
46+ u
6
48+ u
8
410+ etc.
125. If we thus set
1+ 122+ 132+ 142+ etc.= a 1+ 1
28+ 1
38+ 1
48+ etc.= d
1+ 124+ 134+ 144+ etc.= b 1+ 1
210+ 1
310+ 1
410+ etc.= e
1+ 126+ 136+ 146+ etc.= c 1+ 1
212+ 1
312+ 1
412+ etc.= f
etc.,
then the series above is transformed into
a+ bu2 + cu4 + du6 + eu8 + fu10 + etc. =12uu
pi2u
cotpiu.
Now in 118 we found that for the letters A, B, C, D etc., when one sets
s =1uAuBu3 Cu5 Du7 Eu9 etc.
13
one has s = 12 cot12u, and thus, when one replaces
12u by piu, or u by 2piu, one
obtains12cotpiu =
12piu
2Apiu 23Bpi3u3 25Cpi5u5 27Dpi7u7 etc.,and multiplying by piu yields
pi
2ucotpiu =
12uu
2Api2 23Bpi4u2 25Cpi6u4 27Dpi8u6 etc.,from which follows
12uu
pi2u
cotpiu = 2Api2 + 23Bpi4u2 + 25Cpi6u4 + 27Dpi8u6 + etc.
Since we already found that
12uu
pi2u
cotpiu = a+ bu2 + cu4 + du6 + etc.,
it necessarily follows that
a=2 Api2 = 2123 pi2 = 2A12 pi
2
b=23 Bpi4 = 23
12345 pi4 = 2
3B
1234 pi4
c=25 Cpi6 = 25
1237 pi6 = 2
5C
126 pi6
d=27 Dpi8 = 27
1239 pi8 = 2
7D
128 pi8
e=29 Epi10= 29
12311 pi10= 2
9E
1210 pi10
f=211 Fpi12 = 211
12313 pi12= 2
11F
1212 pi12
etc.
[...]
129. From the table of values of the numbers , , , etc. that wecommunicated above in 121, it is apparent that they at first decrease, butthen grow without end. Thus it is worth the effort to investigate in what ratiothese numbers continue to grow, after they reach considerable size. So let bea number far from the beginning in the sequence , , , , etc., and the oneimmediately following. Since the sums of the reciprocal powers are determinedby these numbers, we let 2n be the exponent of the power, in whose sum occurs; 2n+ 2 will be the exponent of the power corresponding to , and n avery large number. Then from 125 one has
1 +122n
+132n
+142n
+ etc. =22n1
1 2 3 (2n+ 1)pi2n,
1 +1
22n+2+
132n+2
+1
42n+2+ etc. =
22n+11 2 3 (2n+ 3)pi
2n+2.
Dividing this series by the former, one finds
1 + 122n+2
+ 132n+2
+ etc.1 + 1
22n+ 1
32n+ etc.
=4pi2
(2n+ 2) (2n+ 3).
14
But because n is a very large number and both series are very closely = 1,
=
(2n+ 2) (2n+ 3)4pi2
=nn
pipi.
Now n indicates which term the number is beyond the first number , andfrom this the number is to the following as pi2 is to n2, and this ratiowould, if n were an infinitely large number, be in complete accordance with thetruth. Because pipi nearly = 10, when one lets n = 100, the hundredth term isapproximately 1000 times smaller than the subsequent one. Thus the numbers, , , etc., and also the Bernoullian A,B, C, D etc., form a highly divergingsequence, which grows more strongly than any geometric sequence of growingterms.
130. Thus if one has found the numbers , , , etc., or A, B, C, Detc., then given a series, whose general term z is a function of its index x, thesummative term Sz can be expressed as follows:
Sz =zdx+
12z +
16 dz1 2dx
130 d
3z
1 2 3 4dx3
+142 d
5z
1 2 3 6dx5 130 d
7z
1 2 3 8dx7
+566 d
9z
1 2 3 10dx9 6912730
d11z
1 2 3 12dx11
+76 d
13z
1 2 3 14dx13 3617510
d15z
1 2 3 16dx15
+43867798
d17z
1 2 3 18dx17 174611330
d19z
1 2 3 20dx19
+854513138
d21z
1 2 3 22dx21 236364091
2730 d
23z
1 2 3 24dx23
+8553103
6 d
25z
1 2 3 26dx25 23749461029
870 d
27z
1 2 3 28dx27
+8615841276005
14322 d
29z
1 2 3 30dx29 etc.
Thus if one knows the integralzdx, or the quantity, whose differential is
= zdx, one finds the summative term by means of continuing differentiation.One must not neglect that a constant value must always be added to thisexpression, of a nature that the sum will = 0, when x becomes 0.
131. If now z is an integral rational function of x, so that the derivativeseventually vanish, then the summative term is represented by a finite expression.We illustrate this by some examples.
First example.Find the summative term of the following series.
15
1 2 3 4 5 x
1+ 9+ 25+ 49+ 81+ + (2x 1)2
Since here z = (2x 1)2 = 4xx 4x+ 1, one haszdx =
43x3 2x2 + x;
because from this, differentiation produces 4xxdx 4xdx+ dx = zdx. Furtherdifferentiation yields
dz
dx= 8x 4, ddz
dx2= 8,
d3z
dx3= 0 etc.
So the summative term sought equals
43x3 2x2 + x+ 2xx 2x+ 1
2+23x 1
3 Const.,
in which the constant must remove the terms 12 13 , so
S (2x 1)2 = 43x3 1
3x =
x
3(2x 1) (2x+ 1) .
So if one sets x = 4, the sum of the first four terms
1 + 9 + 25 + 49 =43 7 9 = 84.
[...]
132. From this general expression for the summative term, the sum for powersof natural numbers, that we communicated in the first part (29 and 61), butwhich we could not prove at that time, follows very easily. Let us set z = xn,so that
zdx = 1n+1x
n+1, and differentiating,
dz
dx= nxn1,
ddz
dx2= n (n 1)xn2, d
3z
dx3= n (n 1) (n 2)xn3,
d5z
dx5= n (n 1) (n 2) (n 3) (n 4)xn5, d
7z
dx7= n (n 1) (n 6)xn7, etc.
16
From this we deduce the following summative term corresponding to the generalterm xn:
Sxn =1
n+ 1xn+1 +
12xn +
16 n2xn1 1
30 n (n 1) (n 2)
2 3 4 xn3
+142 n (n 1) (n 2) (n 3) (n 4)
2 3 4 5 6 xn5
130 n (n 1) (n 6)
2 3 8 xn7
+566 n (n 1) (n 8)
2 3 10 xn9
6912730
n (n 1) (n 10)2 3 12 x
n11
+76 n (n 1) (n 12)
2 3 14 xn13
3617510
n (n 1) (n 14)2 3 16 x
n15
+43867798
n (n 1) (n 16)2 3 18 x
n17
174611330
n (n 1) (n 18)2 3 20 x
n19
+854513138
n (n 1) (n 20)2 3 22 x
n21
2363640912730
n (n 1) (n 22)2 3 24 x
n23
+8553103
6 n (n 1) (n 24)
2 3 . . . 26 xn25
23749461029870
n (n 1) (n 26)2 3 28 x
n27
+8615841276005
14322 n (n 1) (n 28)
2 3 30 xn29
etc.
This expression differs from the former only in that here we have introducedthe Bernoulli numbers A, B, C, D etc., whereas above we used the numbers, , , etc.; the agreement is clear. Thus here we have been able to givethe summative terms for all powers up to the thirtieth, inclusive; if we wantedto perform this investigation via other means, lengthy and tedious calculationswould be necessary.[...]
Part Two, Chapter 6On the summing of progressions via infinite series
140. The general expression, that we found in the previous chapter for thesummative term of a series, whose general term corresponding to the index x
17
is z, namely
Sz =zdx+
12z +
Adz
1 2dx Bd3z
1 2 3 4dx3 +Cd5z
1 2 6dx5 etc.,
actually serves to determine the sums of series, whose general terms are integralrational functions of the index x, because in these cases one eventually arrivesat vanishing differentials. On the other hand, if z is not such a function of x,then the differentials continue without end, and there results an infinite seriesthat expresses the sum of the given series up to and including the term whoseindex = x. The sum of the series, continuing without end, is thus given bytaking x = , and one finds in this way another infinite series equal to theoriginal.
141. If one sets x = 0, the expression represented by the series must vanish,as we already noted; and if this does not occur, one must add to or take awayfrom the sum a constant amount, so that this requirement is satisfied. If this isthe case, then when x = 1 one obtains the first term of the series, when x = 2the sum of the first and second, when x = 3 the sum of the first three terms ofthe series, etc. Because in these cases the sum of the first, first two, first three,etc. terms is known, this is also the value of the infinite series expressing thesum; and thus one is placed in a position to sum countlessly many series.
142. Since when a constant value is added to the sum, so that it vanisheswhen x = 0, the true sum is then found when x is any other number, then itis clear that the true sum must likewise be given, whenever a constant valueis added that produces the true sum in any particular case. Thus suppose itis not obvious, when one sets x = 0, what value the sum assumes and thuswhat constant must be used; one can substitute other values for x, and throughaddition of a constant value obtain a complete expression for the sum. Muchwill become clear from the following.
142a. Consider first the harmonic progression
1 +12+13+14+ + 1
x= s.
Since the general term = 1x , we have z =1x , and the summative term s will
be found as follows. First one haszdx =
dxx = lx; from differentiation one
has
dz
dx= 1
x2,ddz
2dx2=
1x3,d3z
6dx3= 1
x4,
d4z
24dx4=
1x5,
d5z
120dx5= 1
x6, etc.
From this
s = lx+12x
A2x2
+B
4x4 C6x6
+D
8x8 etc.+ Constant.
18
However, the added constant value cannot be determined from the case whenx = 0. So we set x = 1. Since then s = 1, one has
1 =12 A
2+
B
4 C
6+
D
8 etc.+ Constant,
and thus the constant is
=12+
A
2 B
4+
C
6 D
8+ etc.
Consequently the summative term sought is
s= lx+ 12x A2x2 + B4x4 C6x6 + D8x8 etc.+ 12 +A
2 B4 + C6 D8 + etc.
143. Since the Bernoulli numbers A, B, C, D etc. form a diverging series, itis not possible to really know the value of the constant here. But if we substitutea larger number for x, and really find the sum of that many terms, the value ofthe constant can be found easily. Let us set as the end x = 10; the sum of thefirst ten terms
= 2, 928968253968253968,
which must equal the expression for the sum when one sets x = 10 in it, yielding
l10 +120 A200
+B
40000 C6000000
+D
800000000 etc.+ C.
Thus if one substitutes for l10 the hyperbolic logarithm of 10, and in place ofA, B, C etc. substitutes the values found above [122], one obtains for theconstant
C = 0, 5772156649015325,
and this number therefore expresses the sum of the series
12+
A
2 B
4+
C
6 D
8+
E
10 etc.
144. If one substitutes for x a not very large number, then the sum of the[original] series is easy to find, and one obtains the sum
12x A2x2
+B
4x4 C6x6
+D
8x8 etc. = s lx C.
But if x is a very large number, then the sum of this infinite expression can befound in decimal fractions. Now it is clear to begin with that if the [original]series continues infinitely, the sum will have infinite magnitude, because asx =, also lx grows to infinity. But in order nonetheless to be able to give thesum of any number of terms more easily, we express the values of the letters A,
19
B, C etc. in decimal fractions.
A = 0, 1666666666666B = 0, 0333333333333C = 0, 0238095238095D = 0, 0333333333333E = 0, 0757575757575F = 0, 2531135531135G = 1, 1666666666666H = 7, 0921568627451 etc.,
and so
A
2= 0, 0833333333333
B
4= 0, 0083333333333
C
6= 0, 0039682539682
D
8= 0, 0041666666666
E
10= 0, 0075757575757
F
12= 0, 0210927960928
G
14= 0, 0833333333333
H
16= 0, 4432598039216 etc.
First example.Find the sum of one thousand terms of the series 1 + 12 +
13 +
14 +
15+ etc.
20
Set x = 1000, and because
l10 = 2, 3025850929940456840,one has lx = 6, 9077552789821
Const. = 0, 577215664901512x
= 0, 0005000000000
7, 4854709438836
subtr.A
2xx= 0, 0000000833333
7, 4854708605503
addB
4x4= 0, 0000000000000,
thus 7, 4854708605503
is the desired sum of a thousand terms, which is still not seven and a half.[...]
148. After considering the harmonic series we wish to turn to examining theseries of reciprocals of the squares, letting
s = 1 +14+19+
116
+ + 1xx
.
Since the general term of this series is z = 1xx , thenzdx = 1x , the differentials
of z are
dz
2dx= 1
x3,
ddz
2 3dx2 =1x4,
d3z
2 3 4dx3 = 1x5
etc.,
and the sum is
s = C 1x+
12xx
Ax3
+B
x5 Cx7
+D
x9 Ex11
+ etc.,
where the added constant C is determined from one case in which the sum isknown. We therefore wish to set x = 1. Since then s = 1, one has
C = 1 + 1 12+ AB+ CD+ E etc.,
but this series alone does not give the value of C, since it diverges strongly.Above [125] we demonstrated that the sum of the series to infinity is = pipi6 ,and therefore setting x = , and s = pipi6 , we have C = pipi6 , because then allother terms vanish. Thus it follows that
1 + 1 12+ AB+ CD+ Eetc. = pipi
6.
149. If the sum of this series were not known, then one would need todetermine the value of the constant C from another case, in which the sum
21
were actually found. To this aim we set x = 10 and actually add up ten terms,obtaining
s = 1, 549767731166540690.
Further, add 1x = 0, 1
subtr. 12xx = 0, 0051, 644767731166540690
add Ax3
= 0, 0001666666666666661, 644934397833207356
subtr. Bx5
= 0, 0000003333333333331, 644934064499874023
add Cx7
= 0, 0000000023809523811, 644934066880826404
subtr. Dx9
= 0, 0000000000333333331, 644934066847493071
add Ex11
= 0, 0000000000007575751, 644934066848250646
subtr. Fx13
= 0, 0000000000000253111, 644934066848225335
add Gx15
= 0, 000000000000001166
subtr. Hx17
= 711, 644934066848226430 = C.
This number is likewise the value of the expression pipi6 , as one can find bycalculation from the known value of pi. From this it is clear that, although theseries A, B, C, etc. diverges, it nevertheless produces a true sum. [...]
In 150153 Euler explores possible formulas for exact sums of the infiniteseries of reciprocal odd powers of the natural numbers, similar to those hehas already found for the sums of reciprocal even powers in terms of theBernoulli numbers and pi. Using his summation formula he produces highlyaccurate decimal approximations for the sums of reciprocal odd powers allthe way through the fifteenth power, hoping to see a pattern analogous tothe even powers, namely simple fractions times the relevant power of pi. Heis disappointed, however, not to find that they behave similarly to the evenpowers.Then in 154156 Euler uses a sum and the inverse tangent and cotangent
functions to approximate pi to seventeen decimal places with his summationformula, and remarks that it is amazing that one can approximate pi soaccurately with such an easy calculation.
22
157. Now we want to use for z transcendental functions of x, and takez = lx for summing hyperbolic logarithms, from which the ordinary can easilybe recovered, so that
s = l1 + l2 + l3 + l4 + + lx.Because z = lx,
zdx = xlx x,
since its differential is dxlx. Then
dz
dx=
1x,ddz
dx2= 1
x2,
d3z
1 2dx3 =1x3,
d4z
1 2 3dx4 = 1x4,
d5z
1 2 3 4dx5 =1x5, etc.
One concludes that
s = xlx x+ 12lx+
A
1 2x B
3 4x3 +C
5 6x5 D
7 8x7 + etc.+ Const.But for this constant one finds, when one sets x = 1, because then s = l1 = 0,
C = 1 A1 2 +
B
3 4 C
5 6 +D
7 8 etc.,a series that, due to its great divergence, is quite unsuitable even for determiningthe approximate value of C.
158. Nevertheless we can not only approximate the correct value of C, butcan obtain it exactly, by considering Walliss expression for pi provided in theIntroductio [1, vol. 1, chap. 11]. This expression is
pi
2=
2 2 4 4 6 6 8 8 10 10 12 etc.1 3 3 5 5 7 7 9 9 11 11 etc.
Taking logarithms, one obtains from this
lpi l2 = 2l2 + 2l4 + 2l6 + 2l8 + 2l10 + l12 + etc.l1 2l3 2l5 2l7 2l9 2l11 etc.
Setting x = in the assumed series, we have
l1 + l2 + l3 + l4 + + lx=C + (x+ 12) lx x,thus l1 + l2 + l3 + l4 + + l2x=C + (2x+ 12) l2x 2xand l2 + l4 + l6 + l8 + + l2x=C + (x+ 12) lx+ xl2 x,
and therefore l1 + l3 + l5 + l7 + + l (2x 1)= xlx+ (x+ 12) l2 x.Thus because
lpi2 =2l2+ 2l4+ 2l6+ +2l2x l2x 2l1 2l3 2l5 2l (2x 1) ,
23
letting x = yields
lpi
2= 2C + (2x+ 1) lx+ 2xl2 2x l2 lx 2xlx (2x+ 1) l2 + 2x,
and therefore
lpi
2= 2C 2l2, thus 2C = l2piand C = 1
2l2pi,
yielding the decimal fraction representation
C = 0, 9189385332046727417803297,
thus simultaneously the sum of the series
1 A1 2 +
B
3 4 C
5 6 +D
7 8 E
9 10 + etc. =12l2pi.
159. Since we now know the constant C = 12 l2pi, one can exhibit the sum ofany number of logarithms from the series l1 + l2 + l3+ etc. If one sets
s = l1 + l2 + l3 + l4 + + lx,then
s =12l2pi +
(x+
12
)lx x+ A
1 2x B
3 4x3 +C
5 6x5 D
7 8x7 + etc.
if the proposed logarithms are hyperbolic; if however the proposed logarithmsare common, then one must take common logarithms also in the terms 12 l2pi+(x+ 12)lx for l2pi and lx, and multiply the remaining terms
x+ A1 2x
B
3 4x3 + etc.
of the series by 0, 434294481903251827 = n. In this case the common loga-rithms are
lpi = 0, 497149872694133854351268l2 = 0, 301029995663981195213738
l2pi = 0, 79817986835811504956500612l2pi = 0, 399089934179057524782503.
Example.Find the sum of the first thousand common logarithms
s = l1 + l2 + l3 + + l1000.
24
So x = 1000, and
lx= 3, 0000000000000,
and thus xlx=3000, 000000000000012 lx= 1, 5000000000000
12 l2pi= 0, 3990899341790
3001, 8990899341790
subtr. nx= 434, 29448190325182567, 6046080309272.
ThennA12x = 0, 0000361912068
subtr. nB34x3 = 0, 0000000000012
0, 0000361912056
add 2567, 6046080309272the sum sought s=2567, 6046442221328.
Now because s is the logarithm of a product of numbers
1 2 3 4 5 6 1000,it is clear that this product, if one actually multiplies it out, consists of 2568figures, beginning with the figures 4023872, with 2561 subsequent figures.160. By means of this summation of logarithms, one can approximate the
product of any number of factors, that progress in the order of the naturalnumbers. This can be especially helpful for the problem of finding the middle orlargest coefficient of any power in the binomial (a+ b)m, where one notes that,when m is an odd number, one always has two equal middle coefficients, whichtaken together produce the middle coefficient of the next even power. Thussince the largest coefficient of any even power is twice as large as the middlecoefficient of the immediately preceding odd power, it suffices to determine themiddle largest coefficient of an even power. Thus we have m = 2n with middlecoefficient expressed as
2n (2n 1) (2n 2) (2n 3) (n+ 1)1 2 3 4 n .
Setting this = u, one has
u =1 2 3 4 5 2n(1 2 3 4 n)2 ,
and taking logarithms
lu = l1 + l2 + l3 + l4 + l5 + l2n2l1 2l2 2l3 2l4 2l5 2ln.
25
161. The sum of hyperbolic logarithms is
l1 + l2 + l3 + l4 + + l2n = 12l2pi +
(2n+
12
)ln+
(2n+
12
)l2 2n
+A
1 2 2n B
3 4 23n3 +C
5 6 25n5 etc.
and
2l1 + 2l2 + 2l3 + 2l4 + + 2ln= l2pi + (2n+ 1) ln 2n+ 2A
1 2n 2B
3 4n3 +2C
5 6n5 etc.
Subtracting this expression from the former yields
lu = 12lpi 1
2ln+ 2nl2 +
A
1 2 2n B
3 4 23n3 +C
5 6 25n5 etc.
2A1 2n +
2B3 4n3
2C5 6n5 + etc.,
and collecting terms in pairs
lu = l22nnpi
3A1 2 2n +
15B3 4 23n3
63C5 6 25n5 +
255D7 8 27n7 etc.
One has
3A1 2 22n2
15B3 4 24n4 +
63C5 6 26n6
255D7 8 28n8 + etc.
= l(1 +
A
22n2+
B
24n4+
C
26n6+
D
28n8+ etc.
),
so that
lu = l22nnpi
2nl(1 +
A
22n2+
B
24n4+
C
26n6+ etc.
)and thus
u =22n(
1 + A22n2
+ B24n4
+ C26n6
+ etc.)2n
npi.
26
Setting 2n = m,
l(1 + A22n2
+ B24n4
+ C26n6
+ D28n8
+ etc.)
= Am2
+ Bm4
+ Cm6
+ Dm8
+ Em10
+ etc.
A22m4
ABm6
ACm8
ADm10
etc. BB
2m8 BC
m10 etc.
+ A3
3m6+ A
2Bm8
+ A2C
m10+ etc.
+ AB2
m10+ etc.
A44m8
A3Bm10
etc.+ A
5
5m10+ etc.;
and because this expression must equal
3A1 2m2
15B3 4m4 +
63C5 6m6
255D7 8m8 + etc.,
one has
A =3A1 2
B =A2
2 15B
3 4C = AB 1
3A3 +
63C5 6
D = AC +12B2 A2B + 1
4A4 255D
7 8E = AD +BC A2C AB2 +A3B 1
5A5 +
1023E9 10
etc.
162. Now since A = 16 , B =130 , C =
142 , D =
130 , E =
566 , one has
A =14, B = 1
96, C =
27640
, D = 90031211 32 5 7 etc.
Consequently
u =22n(
1 + 124n2
1293n4 +
272135n6 900312193257n8 + etc.
)2nnpi
or
u =22n(1 1
24n2+ 7
293n4 12121335n6 + 1074892193257n8 etc.)2n
npi
,
27
or, if one actually takes the power of the series, approximately
u =22n
npi(1 + 14n +
132n2
1128n3
516128n4 + etc.
) .Thus the middle term in (1 + 1)2n is to the sum 22n of all the terms as
1is tonpi
(1 +
14n
+1
32n2 1128n3
116 128n4 + etc.
);
or, if one abbreviates 4n = , as
1is tonpi
(1 +
1+
122
123
584
+2385
+53166
etc.).
[...]
Second ExampleFind the ratio of the middle term of the binomial (1 + 1)100 to the sum 2100
of all the terms.
For this we wish to use the formula we found first,
lu = l22nnpi
3A1 2 2n +
15B3 4 23n3
63C5 6 25n5 + etc.,
from which, setting 2n = m, in order to obtain the power (1 + 1)m, and aftersubstituting the values of the letters A, B, C, D etc., one has
lu = l2m12mpi
14m
+1
24m3 120m5
+17
112m7 3136m9
+691
88m11 etc.
Since the logarithms here are hyperbolic, one multiplies by
k = 0, 434294481903251,
in order to change to tables, yielding
lu = l2m12mpi
k4m
+k
24m3 k20m5
+17k
112m7 31k36m9
+ etc.,
Now since u is the middle coefficient, the ratio sought is 2m : u, and
l2m
u= l12mpi +
k
4m k24m3
+k
20m5 17k112m7
+31k36m9
691k88m11
+ etc.
Now, since the exponent m = 100,
k
m= 0, 0043429448,
k
m3= 0, 0000004343,
k
m5= 0, 0000000000,
28
yieldingk4m =0, 0010857362k
24m3=0, 0000000181
0, 0010857181 .
Further lpi =0, 4971498726
l 12m =1, 6989700043
l 12mpi =2, 1961198769
l1
2mpi =1, 0980599384k4m k24m3 + etc. =0, 0010857181
1, 0991456565= l 2100
u .
Thus 2100
u = 12, 56451, and the middle term in the expanded power (1 + 1)m
is to the sum of all the terms 2100 as 1 is to 12, 56451.
References
[1] L. Euler, Introduction to analysis of the infinite (transl. John D. Blanton), Springer-Verlag, New York, 1988.
[2] L. Euler, Opera Omnia, B.G. Teubner, Leipzig and Berlin, 1911 .[3] L. Euler, Vollstandige Anleitung zur Differenzial-Rechnung (transl. Johann Michelsen),
Berlin, 1790, reprint of the 1798 edition by LTR-Verlag, Wiesbaden, 1981.[4] L. Euler, Foundations of differential calculus (transl. John D. Blanton), Springer Ver-
lag, New York, 2000.