Upload
virgintebow
View
231
Download
0
Embed Size (px)
Citation preview
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
1/41
1 A Brief Analysis Of The Geometric Series
A Brief
Analysis Of
The Geometric
SeriesType 1 Mathematical
investigation
MD. Rakibur Rahman
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
2/41
2 A Brief Analysis Of The Geometric Series
Introduction
A geometric series is essentially a sequence of real* numbers, where
each term after the first is found by multiplying the first term (t1) by a non-
zero constant known as the common ratio (r).
For example: the sequence 2, 8, 32, 128, is a geometric sequence,
where the first term or t1 is 2, while the common ratio (r) is 4.
While the name first term is self explanatory, the common ratio is
the non-zero constant by which each receding term is multiplied by to obtain
the next. In our example, the common ratio is:
nd
2
st
1
2 term 8= = 4
1 term 2
t
t
Thus, in a more general manner, if the nthterm is tn, then the previous
term is tn-1, then the common ratio or r:
1
n
n
tr
t
Now, the definition of a term of the geometric sequence is the first
term (t1 or a) multiplied by the common ratio to the power of the term
number minus one. To elaborate, for example: if the 1 st term is 5, the common
ratio is 2, then the 4th term will be:
4 1 35 4 = 5 4 = 320
Again, in a more general manner, if the first term is a, the common
ratio is r, then the nth term or tn:
1= nnt a r
This leads us to conclude that the general geometric sequence must be
formulated as shown below:
2 3 1, , , , ... , na a r a r a r a r
*Real numbers are defined as numbers that can be expressed using the real number system, and
thus excludes all imaginary and complex numbers
Sum of a geometric series
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
3/41
3 A Brief Analysis Of The Geometric Series
One of the most important, as well as useful applications/
functions of the geometric series is the sum of the series. If a geometric series
is defined as:
2 3 1, , , , ... , na a r a r a r a r
Then the sum of the series or Snof the first n number of terms is as follows:
2 3 1+ + ... + nnS a a r a r a r a r
For example, 2, 8, 32, 128 is a geometric series. The sum of the first 4 terms of
the series is:
4 2 8 32 128 = 170S
Therefore, the sum of the first 4 terms of the given series is 170.
The sum of a geometric series is one of the most useful
applications of algebra used in our financial world. Interests, loans and
compounding function calculations would not be possible without the use of
geometric sequences. Patterns across natural artifacts occur while following
the rules of a geometric sequence. In physics, rate of change of many functions
occur with respect to a geometric sequence. Chemical reactions at the ionic
levels occur according to a geometric approach. Biological micro-organisms
multiply and breed with respect to a geometric sequence. The importance of a
geometric sequence and the sum is essential to almost any field of education.
Derivation of the sum formula
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
4/41
4 A Brief Analysis Of The Geometric Series
Now, it is relatively simple to calculate the sum of the first 4
numbers, considering the fact only a few terms need to be taken into account
while calculating. However, when the sum of, for example, first 201 terms
need to be calculated, a more effective as well as efficient approach needs to
be considered. Algebra can be used to manipulate the general sequence to
obtain a more feasible equation.
The sum of general geometric sequence is defined as:
2 3 1+ + ... + nnS a a r a r a r a r
Now, the sum this geometric sequence can also be defined as:
1
1
nb
n
b
S a r
Here, the sum of the equation1ba r is found from when b = 1, which gives us
a, to when b = n, which gives us 1nar .
Therefore,
1 2 3 1
1
1 2 3 1
1
= + + ... +
(1 ... ) ---- equation 1
nb n
b
nb n
b
a r a a r a r a r a r
a r a r r r r
Multiplying both sides by r in the second previous statement we get,
1 2 3 4
1
= + + ... +n
b n
b
r a r a r ar a r a r a r
Now, subtracting (n x a) from both sides gives us nnumber of as on each side,
therefore an a for each term in the series:
1 2 3 4
1
1 2 3 4
1
- ( ) = -a+ -a+ -a ... +
- ( ) ( 1) ( 1) ( 1) ( 1) ... ( 1)
nb n
b
nb n
b
r a r n a a r a ar a r a r a r a
r a r n a a r a r a r a r a r
(continued on next page)
(continued from last page)
Now, factoring a on the right side we get,
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
5/41
5 A Brief Analysis Of The Geometric Series
1 2 3 4
1
1 2 3 4
1
1 2 3 4
1
1 2 3 4
1
1 2 3 4
1
- ( ) ( 1 1 1 1 ... 1)
- ( ) ( ... )
( ... ) ( )
( ... )
( ... ) --
nb n
b
nb n
b
n
b n
b
nb n
b
nb n
b
r a r n a a r r r r r
r a r n a a r r r r r n
r a r a r r r r r n n a
r a r a r r r r r n n
r a r a r r r r r -- equa tion 2
Now, subtracting equation 2 from equation 1 we get,
1 1 2 3 1 2 3 4
1 1
1 1 2 2 3 3 1 1
1 1
(1 ... ) - ( ... )
(1 ... ... )
n nb b n n
b b
n nb b n n n
b b
a r r a r a r r r r a r r r r r
a r r a r a r r r r r r r r r
We notice that all of the terms between 1 and rncancels out, thus,
1 1
1 1
1
1
1
1
1
1
(1 )
(1-r) = (1 )
(1 )
(1 )
( 1)
( 1)
n nb b n
b b
nb n
b
nnb
b
nnb
b
a r r a r a r
a r a r
a ra r
r
a ra r
r
Now we remember that
1
1
nb
n
b
S a r (by definition)
Thereby giving us,
( 1)
( 1)
n
n
a rS
r
In order to test the new equation, let us consider the
geometric series:
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
6/41
6 A Brief Analysis Of The Geometric Series
2, 4, 8, 16, 32 and 64
Now, the sum of the first 6 terms of the series by adding the terms manually is:
62 4 8 16 32 64 126S
Now, the first term or a in this series is 2, while the common ratio is 2, and the
number of terms is 6. Using the formula:
6
6
( 1)
( 1)
2(2 1)126
2 1
n
n
a rS
r
S
The sum of the 6 terms given by the derived formula and the manual
calculation is equal.
Therefore, the derived formula is correct and can be used to
calculate the sum of a geometric series.
Convergent geometric series
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
7/41
7 A Brief Analysis Of The Geometric Series
The simplest definition of a convergent geometric series is: A
geometric series whose sum of multiple terms converges, or approaches
towards a certain and real value. Now, convergent geometric series are unique
and therefore, have unique characteristics. There are certain circumstances
under which the sum of a geometric series tends to approach a certain value,
or limit. These circumstances are imposed upon the geometric series by two
contributor, the smaller being the first term or a, while the largest contributor
is the common ratio, or r.
The value of the first term does not have any significant effect on the
converging characteristics of the series. For our example, we will pick a to be
equal to 2.
The value of the common ratio however, greatly contributes to the
convergence of the geometric series.
Fig. 1) Generic number line determined for convergence testing
In the number line of fig. 1, values ranging from and including x, -2,
-1, -0.5, 1
x
, 01
x
, 0.5, 1, 2 and x has been chosen. For the most obvious
reasons, 0 is one of the primary choices made, as it separates positive values
from negative ones. 1 and -1 are also chosen here, because multiplication by
the prior does not change the value of the term, while the latter only causes
sign change. 0.5 and -0.5 are two of the most generic and common fractions or
decimal values used in mathematics, while 1
xand
1
xrepresent general
fractions, where { x > 1|x N}, or x is greater than 1, while being an element
of the natural number system. This allows to test all values up to 0 0}.
If the values of the Sn in table 2 are carefully observed, a pattern can be
observed. The Sn values 2, 4, 6, 8 represent an arithmetic sequence. In this
sequence, the first term (a) is 2, while the common difference (d) is:
2 1 4 2 2d t t
Therefore in this case, the Sn for the geometric sequence represents the tn for
the arithmetic sequence given above.
As we know for an arithmetic sequence:
At t1 = a and common ratio = r,
( 1)nt a n d
In our case, a = 2 and d = 2;
2 ( 1)2
2 2 22
n
n
n
t n
t nt n
Term numbertnfor r = 1
(arn-1)Sn(changes as n increases)
1 2 2
2 2 4
3 2 6
4 2 8
n 2 2n
Table 2) Obtained values oftn and Sn for first term of 2 and common ratio of 1
Here, as n increases, 2n or Sn also increases. Thus, as n , 2nas well. Therefore, at r = 1, the geometric sequence sum is divergent.
a = 2, r = 0.5
Following the method outlined on page 7, we get the values in table 3.
Term number tnfor r = 0.5 Sn(changes as n increases)
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
10/41
10 A Brief Analysis Of The Geometric Series
(arn-1)
( 1)
( 1)
na r
r
1 2 2
2 1 3
3 0.5 3.5
4 0.25 3.75n 22 n
2(2) 4n
Table 3) Obtained values oftn and Sn for first term of 2 and common ratio of
0.5
If we analyze the equation
2(2) 4n , we observe that as n ,
2(2) n 0. Therefore, as
2(2) n 0,
2(2) 4n 4 . Therefore, as the
sum of the geometric series is converging to a certain value, namely 4, we can
conclude that as r = 0.5, the geometric series is convergent.
a = 2, r = 0
Following the method outlined on page 7, we get the values in table 4. .
Term numbertnfor r = 0
(arn-1)
Sn(changes as n increases)
( 1)
( 1)
na r
r
1 2 2
2 0 2
3 0 2
4 0 2
n 0 2
Table 4) Obtained values oftn and Sn for first term of 2 and common ratio of 0
We observe from the above table that as n, Sn=2.
Therefore, the sum of the geometric series converges to the value of
the first term, when the common ratio is 0. In other words, a common ratio of
0 results in a convergent geometric series.
a = 2, r = -2
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
11/41
11 A Brief Analysis Of The Geometric Series
Following the method outlined on page 7 for r = 2, we get the values in
table 5.
Term numbertnfor r = -2
(arn-1)
Sn(changes as n increases)
( 1)
( 1)
na r
r
1 2 2
2 -4 -2
3 8 6
4 -16 -10
n ( 2)n 2 2
( )( 2)3 3
n
Table 5) Obtained values oftn and Sn for first term of 2 and common ratio of -2
In2 2
( )( 2)3 3
n , as n , (-2) , therefore
2 2( )( 2)
3 3
n
. Thus, as the common ratio of a geometric series is -2, the sum of the
series is not convergent (i.e. divergent).
a = 2, r = -1
Following the method outlined on page 8 for r =1, we get the values in
table 6 for r = -1 and a = 2.
Term numbertnfor r = -1
(arn-1)Sn(changes as n increases)
1 2 2
2 -2 0
3 2 2
4 -2 0n 2 ( 1)n 2 if n is odd, 0 if n is even
Table 6) Obtained values oftn and Sn for first term of 2 and common ratio of -1
We can see from the Sn values that as n increases, when n is an odd
number the sum is 2, while when n is odd the sum is 0. Thus, the sum oscillates
between 0 and 2. Therefore, it does not converge or diverge, and is therefore
neither convergent nor divergent.
a = 2, r = -0.5
Following the method outlined on page 9, we get the values in table 7.
Term numbertnfor r = -0.5
(arn-1)Sn(changes as n increases)
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
12/41
12 A Brief Analysis Of The Geometric Series
( 1)
( 1)
na r
r
1 2 2
2 -1 1
3 0.5 1.5
4 -0.25 1.25
n1
4( )2
n
4 1 4( )
3 2 3
n
Table 7) Obtained values oftn and Sn for first term of 2 and common ratio of -
0.5
If we analyze the equation4 1 4
( )3 2 3
n , we see that as n,
1( )
2
n as well. Therefore, this leads to the conclusion that
4 1( )
3 2
n
, and thus the sum of the series converges to4
3at larger values of n.
Therefore, at r = -0.5, the geometric series is convergent.
a = 2, r = x
Following the method outlined on page 7, we get the values in table 8
for a = 2 and r = x.
Term number
tnfor r = x
(arn-1)
Sn(changes as n increases)
( 1)( 1)
n
a rr
1 2 2
2 2x 2+2x
3 2x 2+2x+2x
4 2x3 2+2x+2x2+2x3
n 12xn 2 2
1
nx
x
Table 8) Obtained values oftn and Sn for first term of 2 and common ratio of x
Here, we see that as the value of n approaches larger and larger
values,2 2
1
nx
x
, so does the value of xn. This is because { x > 1|x N}, and
therefore as the value of the exponent increases, so does the value of the
positive natural integer which is greater than 1. Therefore, as n ,
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
13/41
13 A Brief Analysis Of The Geometric Series
2 2
1
nx
x
, for x, given { x > 1|x N}. Therefore, the series is not
convergent for given values of x at { x > 1|x N}.
a = 2, r = -x
Following the method outlined on page 7, we get the values in table 9
for a = 2 and r = -x.
Term numbertnfor r = -x
(arn-1)
Sn(changes as n increases)
( 1)
( 1)
na r
r
1 2 2
2 -2x 2-2x
3 2x 2-2x+2x
4 -2x
2-2x+2x -2x
n 12( )nx 2( ) 2
1
nx
x
Table 9) Obtained values oftn and Sn for first term of 2 and common ratio of -x
Analysis of the equation2( ) 2
1
nx
x
shows us that as n, (-x)n .
This is because when n is odd, the value of (-x)n becomes -xn, and after being
divided by the negative denominator, the sum approaches positive infinity. The
scenario is the opposite for when n is even, and thus the sum approachesnegative infinity. However, this is insignificant, because for neither case the
geometric series converges to a certain value. Therefore, for a common ratio
ofx, the geometric series is divergent*.
*Divergent stands for not convergent, or notconverging towards a certain value.
a = 2, r =1
x
Following the method outline in the previous pages, we obtain the
values in table 10.
Term number tnfor r =1
x Sn(changes as n increases)
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
14/41
14 A Brief Analysis Of The Geometric Series
(arn-1)
( 1)
( 1)
na r
r
1 2 2
22
x2+
2
x
32
2
x 2+
2
x+
2
2
x
43
2
x 2+
2
x+
2
2
x+
3
2
x
n 11
2( )n
x
12( ) 2
11
n
x
x
Table 10) Obtained values oftn and Sn for first term of 2 and common ratio of
1
x
In the equation,
12( ) 2
11
n
x
x
, as n , the value of (1
x)n0. This is because
of the fact that { x > 1|x N}. Thus, as (1
x)n0,
12( )n
x0, and therefore
12( ) 2
11
n
x
x
2 2
1 11
x
or x
x
. Therefore, as the common ratio of a
geometric series is1
x, the series will converge to
2
1
x
x
as n approaches
infinity, given: { x > 1|x N}.
a = 2, r = -1
x
Following the method outline in the previous pages, we obtain the
values in table 11.
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
15/41
15 A Brief Analysis Of The Geometric Series
Term numbertnfor r = -
1
x
(arn-1)
Sn(changes as n increases)
( 1)
( 1)
na r
r
1 2 2
2-
2
x 2-
2
x
32
2
x 2-
2
x+
2
2
x
4 -3
2
x 2-
2
x+
2
2
x-
3
2
x
n112( )n
x
12( ) 2
11
n
x
x
Table 11) Obtained values oftn and Sn for first term of 2 and common ratio of -
1
x
In the equation:
12( ) 2
11
n
x
x
, as n ,1
( )n
x 0. This is because
as x is greater than 1 and is a natural number and is in the denominator, as its
exponents value increases, then value of1
x decreases. As the exponent
approaches ,1
x approaches zero. Therefore, the value of
12( )n
x
approaches zero. This also implies that
12( ) 2
11
n
x
x
2
1
x
x, as n
.Therefore, when1
x is the common ratio of a geometric series, the series
tends to converge to 21
xx
, given { x > 1|x N}.
Results
Now, to sum up our results, table 12 is constructed below.
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
16/41
16 A Brief Analysis Of The Geometric Series
Common ratio (r) Nature of geometric series2 divergent
1 divergent
0.5 convergent
0 convergent
-2 divergent
-1 oscillatory-0.5 convergent
x divergent
-x divergent
1
x convergent
1
x convergent
Table 12) Analyzed common ratios and their respective geometric series nature
We notice from the above table that only 0, 0.5, -0.5,1
x and
1
xproduce geometric series which are convergent. All other values of common
ratio produce divergent series.
Now, it had been previously specified that x > 1. This range
ensures that the value of1
xproduces a value which is less than 1. We notice
from our chart that when r is equal to 1, the series in no longer convergent,
but for values of r < 1; the series is convergent (i.e. at1
x, 0, 0.5, -0.5 and
1
x ) up to -1< r. Again, we notice that as r is equal to -1, the series is not
convergent, but for values of r > -1, the series is convergent. This leads us to
conclude the range of the acceptable r values on a number line (in fig. 2), in
order to obtain a convergent geometric series.
-x -2 -1 -0.5
1
x
0 1
x0.51 2 x
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
17/41
17 A Brief Analysis Of The Geometric Series
Fig. 2) The range of usable common ratios in order to obtain a geometric series
From the above figure and analyzed data we can therefore conclude
that, in order for a convergent geometric series to form
-1 < r < 1
Or, the common ratio is greater than -1 and less than 1.
Derivation of infinite sum formula
Now, we understand that in order to calculate the sum of an infinitely
continuous geometric series, the series must be convergent. Also, we know
that in order for any geometric series to be convergent, the common ratio must
be greater than -1 and less than 1.
Here, by definition:
0
b
b
S a r
In other words, the sum of an infinite series is the sum of all values
from t1 to tb or tn (here b is used instead of n for future algebraic manipulation),
where the value of b approaches infinity.
Now,
0
0
lim
b
b
nb
nb
S a r
S a r
Slight algebraic manipulation shows us that the prior equation is equal
to the latter, because in both the sum of the equation approaches infinity.
Only in the second one, the value of b approaches n, while the value of n
approaches infinity (making both equations equal).
0
limn
b
nb
S a r
Now by definition, we know that:
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
18/41
18 A Brief Analysis Of The Geometric Series
0
nb
b
a r
= nS
Now,
If
( 1)( 1)
n
n a rSr
Then,
0
nb
b
a r
=( 1)
( 1)
na r
r
Thus substituting this in0
limn
b
nb
S a r
we get,
(ac co rding to p roperties of limits)
( 1)lim
( 1)
lim - lim( 1) ( 1)
lim lim( 1) ( 1)
n
n
n
n n
n
n n
a rS
r
a a rS
r r
a r aS
r r
Now, we know that -1 < r < 1; thus when n , rn0. Thus, as rn 0,
( 1)
na r
r 0.
Thus,
(as the value of n doe s not have a ny effect o n this equa tion, the limit ca n be remo ved)
= 0 - lim( 1)
=-( 1)
= where -1
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
19/41
19 A Brief Analysis Of The Geometric Series
Verification of formula
Now, from the previous section we understand that at common ratio = -
0.5, the geometric series is convergent, where the sum of the geometric series
converges to4
3(as evaluated algebraically on page 12). Now, using our
derived formula at =1
aS
r
, let us verify the geometric series where a = 2, r =
-0.5.
Here,
2=
1 0.5
4=
3
S
S
Thus, as both algebraic manipulation and our derived formula gives us
the sum of infinity as4
3, we can verify the formula to be valid and usable in
future cases.
Interpretation of transformation for given equations
Visualization of an idea or fact is how human beings analyze critical
data as if it were simple. This is the function of a graph. Simply stated, a graph
is generally an equation plotted on multiple axes in order to represent theequation visually, for critical analysis. Although this may be done using algebra
and/or limits, it is much easier to understand the equation visually first, and
then evaluate or confirm using algebra. We will analyze some equations now
and later interpret to how this relates to a geometric series.
Equation set 1
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
20/41
20 A Brief Analysis Of The Geometric Series
In the equation set 1:
1
2
3
2 -- equation 1
2 1 --equa tion 2
3(2 1) --equat ion 3
x
x
x
y
y
y
Equation 1 undergoes a set of transformations in order to become equation 3.
Fig. 3) shows the three of these equations plotted on the same set of axis. Note
that these equations are drawn to represent the equations only, and not the
geometric series which they are meant to represent later on.
A ti-83 Plus graphing calculator may be used for plotting these graphs
for ease of analysis. The following range of variables may be used:
x: [0,9, 1]
y: [0, 12 , 1]
A screenshot* of the window is given as below:
*An artificially computer generated mod of the Ti-83+ was used via ROM dump
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
21/41
21 A Brief Analysis Of The Geometric Series
Now, the following graphs are plotted on y1, y2, y3 as shown:
Now, we get the graph as given below (graph also included in fig. 3):
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
22/41
22 A Brief Analysis Of The Geometric Series
In general sense, a graphical transformation is when a set of changes
are made to an original equation in order to obtain a different and unique
equation. Let us consider the equation set 1 as listed above to discuss the
transformations applied there.
Referring to fig. 3), we see that Equation 1, which is 1 2xy , is an
exponential curve which intersects the y-axis through the point (0,1). Now, the
graph is moved down (vertically translated) by 1 unit on the y-axis in order to
obtain equation 2. We can see this algebraically as well as equation 2 is
2 2 1xy , which is essentially 1 subtracted from equation 1. As we can
observe in the figure, equation 2 intersects the y axis at (0,0), which is 1 unit
below the intersection of equation 1.
Equation 3 is essentially equation 2 with a vertical stretch factor of 3. In
Lehmans terms, equation 3 has been stretched 3 units on a vertical aspect
from equation 2. This can be observed visually in figure 3, as well as
mathematically, as equation 3 is 3 3(2 1)xy , which is basically 3 times
2 2 1xy .
On a visual terrain:
1 2xy 2 2 1
xy 3 3(2 1)xy
Equation set 2
Equation set 2 utilizes the same set of skills used in equation set 1.
Equation set 2:
1
2
3
1( ) -- eq ua tion 1
2
1( ) 1 --equa tion 2
2
13[( ) 1] --eq ua tion 3
2
x
x
x
y
y
y
Vertically translated by -1 unit Vertical stretch factor of 3 applied
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
23/41
23 A Brief Analysis Of The Geometric Series
For equation set 2, the range of x and y are as follows:
x: [0, 9, 1)
y: [-3, 1, 1]
The window (which directly corresponds with the range) is as follows:
The graph obtained (also illustrated in fig. 4) is as follows:
Referring to fig. 4, we observe that as x values approaches infinity, the y
values of the equations y1, y2, y3 respectively approach 0, -1 and -3.
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
24/41
24 A Brief Analysis Of The Geometric Series
From equation 1 or 11
( )2
xy , the equation 21
( ) 12
xy was moved down
1 unit or vertically translated 1 unit down. The vertical asymptote for equation
1, which is y = 0, was transformed into y = -1 for equation 2, which was also
vertically translated 1 unit down.
From equation 2 to 3, 21
( ) 12
xy or equation 2 was
transformed with a vertical stretch factor of 3 with respect to the y-axis. Inthis case, the vertical asymptote was transformed from y =-1 to y = -3, which is
a vertical transformation with a stretch factor of 3 as well.
1
1( )2
xy 21
( ) 12
xy 31
3[( ) 1]2
xy
Vertically translated by -1 unit Vertical stretch factor of 3 applied
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
25/41
25 A Brief Analysis Of The Geometric Series
Equation set 3
Although equation set 3 looks very similar to equation set 2, it is
extremely different when we graph it.
Equation set 3 is defined as follows:
1
2
3
1( )
2
1( ) 1
2
13[( ) 1]
2
x
x
x
y
y
y
We will be using the same windows as equation set 3 to graph this equation set.
After applying the values inside the calculator, we obtain the following graph:
This of course, from a direct perspective, signifies nothing but a
blank graph. However, if we look at it in an algebraic approach initially, and
then a graphical, we will understand the problem.
Our function consists of an exponential graph with a negative
base. Let us take 11
( )2
xy as an example. If for example, x was equal to
natural numbers 0, 1 and 2, the corresponding y values would be 1, -0.5 and
0.25 respectively, using simple substitution method.
Now, let us consider non-natural numbers such as1
2. Using the
real numbers system, 11
( )2
xy cannot be evaluated at x =1
2, since it
consists of taking the second root of a negative number, which cannot be
accomplished without the use of complex numbers.
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
26/41
26 A Brief Analysis Of The Geometric Series
This means that using a real number system, equation set 3
cannot be graphed continuously, which results in a discontinuous function. Of
course, this can be graphed continuously using polar co-ordinates on a three
dimensional plane, however since this increases the complexity of this report
and is not included in the International Baccalaureate Standard Level
curriculum, we will not be going into details here.
Therefore we can conclude that, using the real number co-
ordination graphing system, equation set 3, which essentially consists of
exponential equations with negative bases, is only defined at:
x:[x N]*
This is verified by using the trace function of the ti-83+ and
placing x values which only consist of the natural number system, thus giving us
responding y co-ordinates.
Now, graphing the defined points on the same window as
equation set 2 in fig. 5) shows us that the points on the graphs are transformedaccording to the following measures:
1
1(- )
2
xy 2
1(- ) 1
2
xy
3
13[(- ) 1]
2
xy
The equation set 3 has the same values for vertical asymptotes as set 2.
*Hint: Incidentally, the sum of a geometric series is also defined at n:[n N]; but we will be goinginto details later!
Vertically translated by -1 unit Vertical stretch factor of 3 applied
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
27/41
27 A Brief Analysis Of The Geometric Series
Equation set 4
Equation set 4 is, as discussed above, only defined at x:[x N].
Equation set 4 is defined as below:
1
2
3
( 2)
( 2) 1
3[( 2) 1]
x
x
x
y
y
y
Again, graphing on the ti-83+ does not provide any useful visual
information. Therefore, we will be using the same graphing method as used in
fig. 5); i.e. by tracing the graph at x N, and then plotting the points in fig. 6).
Our two variables x and y will be defined as:
x: [x N|0, 3, 1]
y: [-12, 12, 1]
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
28/41
28 A Brief Analysis Of The Geometric Series
The equation set 4 has been transformed by the following
measures, as observations from figure 6 and algebraic characteristics dictate:
1 (-2)xy 2 (-2) 1
xy 3 3((-2) 1)xy
The given equations do not approach or converge to a certain
value, but extend towards positive and negative infinity on both axes.
Therefore, they do not have any vertical or horizontal asymptotes.
Vertically translated by -1 unit Vertical stretch factor of 3 applied
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
29/41
29 A Brief Analysis Of The Geometric Series
Rewriting final equation of set 1 in terms of Sum of geometric series formula
The final equation given in equation set 1 is 3y , where:
3 3(2 1)x
y
Now, our purpose is to rewrite this equation in terms of the equation derived
for the sum of the geometric series, or:
( 1)
( 1)
n
n
a rS
r
Now,
3
3
3(2 1)
13(2 1)
2 1
x
x
y
y
This resembles the arbitrary equation
( 1)
( 1)
n
n
a rS
r
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
30/41
30 A Brief Analysis Of The Geometric Series
Where, a = 3, r = 2, x = n, nS = 3y .
This allows us to conclude that equation 3 of equation set 1
represents the sum of a geometric series, where the variables are defined as a
= 3, r = 2, x = n, nS = 3y .
Now as we know, a geometric series is defined generally as follows:
2 3 1, , , , ... , na a r a r a r a r
In this case, we know that a = 3 and r = 2,
Thus the series represented by equation set 1s final equation is:
2 13,3 (2), 3 (2) ,..., 3 (2)n
Rewriting final equation of set 2 in terms of Sum of geometric series formula
The final equation given in equation set 2 is 3y , where:
3
13[( ) 1]
2
xy
Using algebraic manipulation,
3
3
13[( ) 1]
2
1-0.5{3[( ) 1]}
20.5
x
x
y
y
3
1-1.5[( ) 1]}
21
12
x
y
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
31/41
31 A Brief Analysis Of The Geometric Series
In terms of
( 1)
( 1)
n
n
a rS
r
Where a = -1.5, r =1
2, x = n, nS = 3y
This allows us to conclude that equation 3 of equation set 2 represents the sum
of a geometric series, where the variables are defined as
a = -1.5, r =1
2, x = n, nS = 3y .
Now as we know, a geometric series is defined generally as follows:
2 3 1, , , , ... , na a r a r a r a r
In this case, we know that a = -1.5 and r = 12
,
Thus the series represented by equation set 2s final equation is:
2 11 1 11.5, 1.5 ( ), -1.5 ( ) ,..., -1.5 ( )2 2 2
n
Rewriting final equation of set 3 in terms of Sum of geometric series formula
The final equation given in equation set 3 is 3y , where:
3
13[(- ) 1]
2
xy
Now,
3
3
3
13[(- ) 1]
2
1-1.5{3[(- ) 1]}
2 1.5
1-4.5[(- ) 1]}
21
12
x
x
x
y
y
y
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
32/41
32 A Brief Analysis Of The Geometric Series
Which is in terms of
( 1)
( 1)
n
n
a rS
r
3
1Where a = -4.5, r = - , x = n and y
2n
S
Now as we know, a geometric series is defined generally as follows:
2 3 1, , , , ... , na a r a r a r a r
In this case, we know that a = -4.5 and r =1
2 ,
Thus the series represented by equation set 3s final equation is:
2 11 1 14.5, 4.5 (- ), -4.5 (- ) ,..., -4.5 (- )
2 2 2
n
Rewriting final equation of set 4 in terms of Sum of geometric series formula
The final equation given in equation set 3 is 3y , where:
3 3((-2) 1)xy
Here,
3
3
3
3((-2) 1)
-3{3((-2) 1)}
3
-9((-2) 1)}
2 1
x
x
x
y
y
y
Which is in the form
( 1)
( 1)
n
n
a rS
r,
3Where, a 9, r 2,x = n and y
nS
Now as we know, a geometric series is defined generally as follows:
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
33/41
33 A Brief Analysis Of The Geometric Series
2 3 1, , , , ... , na a r a r a r a r
In this case, we know that a = -9 and r = -2,
Thus the series represented by equation set 4s final equation is:
2 19, 9 (-2), -9 (-2) ,..., -9 (-2)n
Convergence and divergence of final equation in the 4 sets
To define convergence of a graph, it is essentially when as the x
co-ordinates increases or decreases infinitely, the y co-ordinates reach a
certain, real number value. For example, in the graph y = x2, as x approaches
positive infinity, the y-values also approach positive infinity. Therefore, y = x2
is not convergent. While on the other hand, in the equation1
yx
, as x
approaches positive infinity, y approaches 0. Therefore,1
yx
is converging
towards y = 0.
Convergence of final equation in set 1
The final equation in equation set 1 is defined as:
3 3(2 1)xy
Which can be rewritten as,
3
3(2 1)
2 1
x
y
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
34/41
34 A Brief Analysis Of The Geometric Series
Thereby representing the geometric series
2 13,3 (2), 3 (2) ,..., 3 (2)n
Now if we look at 3 3(2 1)xy algebraically, as x approaches ,
3
3
3
lim lim 3(2 1)
lim lim 3(2) lim( 3)
lim not def ined, or y also approaches infinity
x
x x
x
x x x
x
y
y
y
We can observe this in fig. 3), where we see that as x-approaches infinity, y-
also approaches infinity.
Now, in terms of the geometric sum formula, let us consider the
sequence:
2 13,3 (2), 3 (2) ,..., 3 (2)n
( 1)
( 1)
n
n
a rS
r
Here a = 3, r =2
3(2 1)
(2 1)
,
lim lim 3(2 1)
lim do es not exist, or ap proac hes po sitive infinity.
n
n
n
nn n
nn
S
Now
S
S
In addition, if we try to apply our infinite geometric sum
formula:
= where -1
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
35/41
35 A Brief Analysis Of The Geometric Series
Therefore, in the equation 3 3(2 1)x
y as well as the geometric series
represented by the equation, the y-values and the sum of geometric series
value is not convergent.
Convergence of final equation in set 2
3
13[( ) 1]
2
xy is the final equation in set 2, and represents the
geometric series,2 11 1 11.5, 1.5 ( ), -1.5 ( ) ,..., -1.5 ( )
2 2 2
n
Observations from fig. 4) show us that as x approaches positive infinity,
the y-values approach -3.
Algebraically,
3
3
3
3
3
13[( ) 1]
2
1lim lim 3[( ) 1]
2
1lim 3lim ( ) lim 3
2
lim 0 3
lim 3
x
x
x x
x
x x x
x
x
y
y
y
y
y
Thus, as x approaches positive infinity in 313[( ) 1]2
xy , the y
values approach -3.
Using the geometric series2 11 1 11.5, 1.5 ( ), -1.5 ( ) ,..., -1.5 ( )
2 2 2
n ,
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
36/41
36 A Brief Analysis Of The Geometric Series
( 1)
( 1)
n
n
a rS
r
Where a = -1.5, r =1
2
1-1.5(( ) 1)
2
1( 1)2
13( ) 3
2
1lim 3lim( ) lim 3
2
lim 0 3
lim 3
n
n
n
n
n
nn n n
n
n
nn
S
S
S
S
S
Now, using the derived formula:
= where -1
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
37/41
37 A Brief Analysis Of The Geometric Series
3
13[(- ) 1]
2
xy is the final equation in set 3, and we observe
that it is a discontinuous function, only defined at x:[x N]. This is also the
same scenario with its geometric series, where:
2 11 1 14.5, 4.5 (- ), -4.5 (- ) ,..., -4.5 (- )2 2 2
n , and n:[n N)
Algebraically,
3
3
3
3
13[(- ) 1]
2
1lim 3lim(- ) lim 3
2
lim 0 3
lim 3
x
x
x x x
x
x
y
y
y
y
Thus, as x approaches positive infinity, the y-values approach -3.
If we look at fig. 5) we observe that as the x co-ordinates
approach positive infinity, the y-values approach -3 as well.
In the geometric series,
2 11 1 14.5, 4.5 (- ), -4.5 (- ) ,..., -4.5 (- )2 2 2
n
The sum of the series represents the equation 31
3[(- ) 1]2
xy
Where,
( 1)
( 1)
n
n
a rS
r
and a = -4.5, r=
1-2
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
38/41
38 A Brief Analysis Of The Geometric Series
1-4.5[(- ) 1]
2
1( 1)
2
1
lim lim{3[(- ) 1]}2
1lim 3lim (- ) lim 3
2
lim 3
n
n
n
nn n
n
nn n n
nn
S
S
S
S
Therefore, as n approaches positive infinity, the sum of the geometric
series approaches -3.
If we examine the series using our infinite sum formula of,
= where -1
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
39/41
39 A Brief Analysis Of The Geometric Series
Algebraically,
3
3
3
3((-2) 1)
lim 3lim( 2) lim 3
lim does no t exist , o r approaches
x
x
x x x
x
y
y
y
Using the geometric sum formula,
( 1)
( 1)
n
n
a rS
r
where a =-9 and r = -2,
-9((-2) 1)
( 2 1)
3(-2) 3
lim 3 lim (-2) lim 3
lim DNE or a pp roa ches
n
n
n
n
nn
n n n
nn
S
S
S
S
If we apply our infinite sum formula here,
= where -1
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
40/41
40 A Brief Analysis Of The Geometric Series
2 11 1 14.5, 4.5 (- ), -4.5 (- ) ,..., -4.5 (- )2 2 2
n
Now, the initial observation made from the above equation, as well as
the other equation sets is that:
The base of the exponent, i.e.1
-2
in this case, represents the
common ratio value in its respective geometric series.
This can be verified using other equations in the sets as well.
The second observation that is made from this report is that,
The value of the first term, or a, is equal to the vertical stretch
factor, i.e. 3 in this case, multiplied by the common ratio subtracted by 1.
In simpler terms: a = (vertical stretch factor) (common ratio -1)
One can also conclude that,
a or t1 = (vertical stretch factor) (base of exponent -1) [as base of
exponent = common ratio]
The third, and most important observation made:
If the base of the exponent is less than 1 or greater than -1, then
the equation, as well as its sum of infinity of the geometric series, will
converge to a real value. On the other hand, if the base of the exponent is
greater than 1 or less than -1, then the equation as well as its sum of
infinity will be divergent.
In general terms,
If zx is the given exponent in y = k(zx-1), Then:
7/29/2019 A Brief Analysis of the Geometric Series UPLOAD VERSION
41/41
if -1