Kohshi Gokita
March 28, 2010
IB Math SL. Per. 8
Internal AssessmentInfinite Summation
The infinite sequence below is considered
t 0=1 t 1=(xlna )01
1 ! t 2=
(xlna )2
2! …………..
So, the general term of this sequence is expressed as follows;
t n=( xlna)n
n!
The sum T n(a,x) of the first (n+1) terms of the above sequence is considered
T n(a ,x) = t 0 + t 1 + t 2 + t 3 + …………….+t n
= ∑k=0
n
tk
=∑k=0
n (xlna)k
k !
Aim;
In this task, the sum of infinite sequence t n is considered, hence, the value of T n
(a, x) as a n approaches∞,
limn → ∞
T n(a , x)
would be calculated
I. In this part of investigation, the value of x is fixed on 1 and the effect of change in the value of a over the value of T n(a, x) would be observed by numerical simulation.
1. Consider the following sequence of terms where x=1 and a=2. And calculate the
sum T n of the first n terms of the above sequence for 0≤n≤10.
1, + ( ln 2 )❑
1 +
( ln 2 )2
2∗1 +
( ln2 )3
3∗2∗1 + ………..
0≤n≤10:
a x n t T
2 1 0 1 1
1 0.693147 1.693147
2 0.240227 1.933374
3 0.055504 1.988878
4 0.009618 1.998496
5 0.001333 1.999829
6 0.000154 1.999983
7 1.53E-05 1.999999
8 1.32E-06 2
9 1.02E-07 2
10 7.05E-09 2
0 2 4 6 8 10 120
0.5
1
1.5
2
2.5
0≤n≤20:
a x n t T
2 1 0 1 1
1 0.693147 1.693147
2 0.240227 1.933374
3 0.055504 1.988878
4 0.009618 1.998496
5 0.001333 1.999829
6 0.000154 1.999983
7 1.53E-05 1.999999
8 1.32E-06 2
9 1.02E-07 2
10 7.05E-09 2
11 4.45E-10 2
12 2.57E-11 2
13 1.37E-12 2
14 6.78E-14 2
15 3.13E-15 2
16 1.36E-16 2
17 5.53E-18 2
18 2.13E-19 2
19 7.77E-21 2
20 2.69E-22 2
0 5 10 15 20 250
0.5
1
1.5
2
2.5
*According to data, as n increases, sum of infinity approaches to the value of a. In this
case, the sum gets closed to 2.
2. Consider the following sequence of terms where x=1 and a=3. And calculate the
sum T n of the first n terms of the above sequence for 0≤n≤10.
1, + ( ln 3 )❑
1 +
(ln 3 )2
2∗1 +
( ln3 )3
3∗2∗1 + ………..
0≤n≤10:
a x n t T
3 1 0 1 1
1 1.098612 2.098612
2 0.603474 2.702087
3 0.220995 2.923082
4 0.060697 2.983779
5 0.013336 2.997115
6 0.002442 2.999557
7 0.000383 2.99994
8 5.26E-05 2.999993
9 6.42E-06 2.999999
10 7.06E-07 3
0 2 4 6 8 10 120
0.5
1
1.5
2
2.5
3
3.5
0≤n≤20:
a x n t T
3 1 0 1 1
1 1.098612 2.098612
2 0.603474 2.702087
3 0.220995 2.923082
4 0.060697 2.983779
5 0.013336 2.997115
6 0.002442 2.999557
7 0.000383 2.99994
8 5.26E-05 2.999993
9 6.42E-06 2.999999
10 7.06E-07 3
11 7.05E-08 3
12 6.45E-09 3
13 5.45E-10 3
14 4.28E-11 3
15 3.13E-12 3
16 2.15E-13 3
17 1.39E-14 3
18 8.49E-16 3
19 4.91E-17 3
20 2.7E-18 3
0 5 10 15 20 250
0.5
1
1.5
2
2.5
3
3.5
*According to data, as n increases, sum of infinity approaches to the value of a. In this
case, the sum gets closed to 3.
3. Consider the following sequence of terms where x=1 and a=4. And calculate the
sum T n of the first n terms of the above sequence for 0≤n≤10.
1, + (ln 4 )❑
1 +
( ln 4 )2
2∗1 +
(ln 4 )3
3∗2∗1 + ………..
0≤n≤10:
a x n t T
4 1 0 1 1
1 1.386294 2.386294
2 0.960906 3.3472
3 0.444033 3.791233
4 0.15389 3.945123
5 0.042667 3.987791
6 0.009858 3.997649
7 0.001952 3.999601
8 0.000338 3.99994
9 5.21E-05 3.999992
10 7.22E-06 3.999999
0 2 4 6 8 10 120
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Series2
0≤n≤20:
a x n t T
4 1 0 1 1
1 1.386294 2.386294
2 0.960906 3.3472
3 0.444033 3.791233
4 0.15389 3.945123
5 0.042667 3.987791
6 0.009858 3.997649
7 0.001952 3.999601
8 0.000338 3.99994
9 5.21E-05 3.999992
10 7.22E-06 3.999999
11 9.1E-07 4
12 1.05E-07 4
13 1.12E-08 4
14 1.11E-09 4
15 1.03E-10 4
16 8.89E-12 4
17 7.25E-13 4
18 5.59E-14 4
19 4.08E-15 4
20 2.82E-16 4
0 5 10 15 20 250
0.5
1
1.5
2
2.5
3
3.5
4
4.5
*According to data, as n increases, sum of infinity approaches to the value of a. In this
case, the sum gets closed to 4.
4. Now, consider a general sequence where x=1. And calculate the sum T n of the
first n terms of the above sequence for 0≤n≤10 for different values of a.
1, + ( lna)❑
1 +
( lna)2
2∗1 +
(lna )3
3∗2∗1 + ………..
If x=1 and a=10, the equation will be
1, + (ln 10 )❑
1 +
( ln 10 )2
2∗1 +
( ln10 )3
3∗2∗1 + ………..
0≤n≤10:
a x n t T
10 1 0 1 1
1 2.302585 3.302585
2 2.650949 5.953534
3 2.034679 7.988213
4 1.171255 9.159468
5 0.539383 9.698851
6 0.206996 9.905847
7 0.068089 9.973936
8 0.019598 9.993534
9 0.005014 9.998548
10 0.001154 9.999702
0 2 4 6 8 10 120
2
4
6
8
10
12
0≤n≤20:
a x n t T
10 1 0 1 1
1 2.302585 3.302585
2 2.650949 5.953534
3 2.034679 7.988213
4 1.171255 9.159468
5 0.539383 9.698851
6 0.206996 9.905847
7 0.068089 9.973936
8 0.019598 9.993534
9 0.005014 9.998548
10 0.001154 9.999702
11 0.000242 9.999944
12 4.64E-05 9.99999
13 8.21E-06 9.999998
14 1.35E-06 10
15 2.07E-07 10
16 2.98E-08 10
17 4.04E-09 10
18 5.17E-10 10
19 6.27E-11 10
20 7.21E-12 10
0 5 10 15 20 250
2
4
6
8
10
12
*According to data, as n increases, sum of infinity approaches to the value of a. In this
case, the sum gets closed to 10.
According to whole observation of experiment, it is proved that as n increases, the
sum approaches to value of a. In addition, as a increases, the converge speed gets slow
and slow. In cases of a=2 and x=1, sum becomes 2 when n is 8. However, in case of
a=10 and x=1, sum becomes 10 when n is 14. Ultimately, the general statement of the
infinite summation is
∑n=0
∞ ( lna)n
n!=a
II. In this part of investigation, how the change in the positive value of x as well as the positive value of a would affect the value of T n(a, x) would be observed by numerical simulation.
1. Consider the following sequence of terms where x=5 and a=2. And calculate the
sum T n of the first n terms of the above sequence for 0≤n≤10.
1, + (5 ln 2 )❑
1 +
(5 ln 2 )2
2∗1 +
(5 ln 2 )3
3∗2∗1 + ………..
0≤n≤10:
a x n t T
2 5 0 1 1
1 3.465736 4.465736
2 6.005663 10.4714
3 6.938014 17.40941
4 6.011331 23.42074
5 4.166737 27.58748
6 2.406802 29.99428
7 1.19162 31.1859
8 0.51623 31.70213
9 0.198791 31.90092
10 0.068896 31.96982
0 2 4 6 8 10 120
5
10
15
20
25
30
35
0≤n≤20;
a x n t T
2 5 0 1 1
1 3.465736 4.465736
2 6.005663 10.4714
3 6.938014 17.40941
4 6.011331 23.42074
5 4.166737 27.58748
6 2.406802 29.99428
7 1.19162 31.1859
8 0.51623 31.70213
9 0.198791 31.90092
10 0.068896 31.96982
11 0.021707 31.99152
12 0.006269 31.99779
13 0.001671 31.99946
14 0.000414 31.99988
15 9.56E-05 31.99997
16 2.07E-05 31.99999
17 4.22E-06 32
18 8.13E-07 32
19 1.48E-07 32
20 2.57E-08 32
0 5 10 15 20 250
5
10
15
20
25
30
35
*According to data, the graph’s shape changes. In this case, sum becomes 32 when n is
16. In addition, in the graph, there is a inflection point. The curve of the graph changes.
a=2, x=5 limn → ∞
T n(2 ,5) = 25 = 32
2. Consider the following sequence of terms where x=3 and a=2. And calculate the
sum T n of the first n terms of the above sequence for 0≤n≤10.
1, + (3 ln 2 )❑
1 +
(3 ln 2 )2
2∗1 +
(3 ln 2 )3
3∗2∗1 + ………..
0≤n≤10:
a x n t T
2 3 0 1 1
12.0794
42
3.0794
42
22.1620
39
5.2414
8
31.4986
11
6.7400
91
40.7790
68
7.5191
59
50.3240
05
7.8431
65
60.1122
92
7.9554
57
70.0333
58
7.9888
14
80.0086
71
7.9974
85
90.0020
03
7.9994
88
100.0004
17
7.9999
05
0 2 4 6 8 10 120
1
2
3
4
5
6
7
8
9
0≤n≤20:
a x n t T
2 3 0 1 1
12.0794
42
3.0794
42
22.1620
39
5.2414
8
31.4986
11
6.7400
91
40.7790
68
7.5191
59
50.3240
05
7.8431
65
60.1122
92
7.9554
57
70.0333
58
7.9888
14
80.0086
71
7.9974
85
90.0020
03
7.9994
88
100.0004
17
7.9999
05
117.88E-
05
7.9999
84
121.36E-
05
7.9999
97
132.18E-
068
143.24E-
078
154.49E-
088
165.84E-
098
17 7.15E- 8
10
188.25E-
118
199.03E-
128
209.39E-
138
0 5 10 15 20 250
1
2
3
4
5
6
7
8
9
*According to data, the graph’s shape changes. In this case, sum becomes 8 when n is
about5. In addition, in the graph, there is a inflection point. The curve of the graph
changes.
a=2, x=3 limn → ∞
T n(2 ,3) = 23 = 8
According to whole observation of investigation, it is proved that if the value of x
changes, the sum also changes. However, it is not that the sum approaches value of a.
The sum approaches the number of ax. In investigation in part I, the sum becomes the
same value as a but it is that because x is fixed on 1. In the case of a=2 and x=5, the
total sum becomes 25, 32. Ultimately, the general statement of the infinite summation is
∑n=0
∞ ( lna)n
n!=ax
III. In this part of investigation, how the negative value of x as well as the positive value of a would affect the value of T n(a, x) would be observed by numerical simulation. In addition, how the change in the positive value of a as well as the value of x would affect the value of T n(a, x) would be observed by numerical simulation.
1. Consider the following sequence of terms where x= -2 and a=2. And calculate the
sum T n of the first n terms of the above sequence for 0≤n≤10.
1, + (−2 ln 2 )❑
1 +
(−2 ln 2 )2
2∗1 +
(−2 ln 2 )3
3∗2∗1 + ………..
0≤ n ≤10
a x n t T
2 -2 0 1 1
1-1.38629-0.38629
20.9609060.574612
3-0.444030.130579
4 0.153890.284469
5-0.042670.241801
60.009858 0.25166
7-0.001950.249707
80.0003380.250046
9-5.2E-050.249994
107.22E-060.250001
0 2 4 6 8 10 12
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Series2
0≤ n ≤20
a x n t T
2 -2 0 1 1
1 -1.38629 -0.38629
2 0.960906 0.574612
3 -0.44403 0.130579
4 0.15389 0.284469
5 -0.04267 0.241801
6 0.009858 0.25166
7 -0.00195 0.249707
8 0.000338 0.250046
9 -5.2E-05 0.249994
10 7.22E-06 0.250001
11 -9.1E-07 0.25
12 1.05E-07 0.25
13 -1.1E-08 0.25
14 1.11E-09 0.25
15 -1E-10 0.25
16 8.89E-12 0.25
17 -7.3E-13 0.25
18 5.59E-14 0.25
19 -4.1E-15 0.25
20 2.82E-16 0.25
0 5 10 15 20 25
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Series2
*According to the graph, the shape has dramatically changed from the graph of 0≤ x.
When n is 2, the sum value becomes negative but when n is 3, the value goes to positive
again. However, when n is 4, the sum value does not go to negative but it decreases
once again. From n becomes 4, the sum value constantly becomes 0.25.
a=2, x=-2 limn → ∞
T n(2 ,−2) = 2−2 = 0.25
2. Consider the following sequence of terms where x= -2 and a= 1/2. And calculate the
sum T n of the first n terms of the above sequence for 0≤n≤10.
1, + (−2 ln ( 12))
❑
1 + (−2 ln ( 1
2))
2
2∗1
+ (−2 ln ( 12))
3
3∗2∗1
+ ………..
0≤n≤10
a x n t T
0.5 -2 0 1 1
11.3862942.386294
20.960906 3.3472
30.4440333.791233
4 0.153893.945123
50.0426673.987791
60.0098583.997649
70.0019523.999601
80.000338 3.99994
95.21E-053.999992
107.22E-063.999999
0 2 4 6 8 10 120
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Series2
0≤n≤20a x n t T
2 -2 0 1 1
1 -1.38629 -0.38629
2 0.960906 0.574612
3 -0.44403 0.130579
4 0.15389 0.284469
5 -0.04267 0.241801
6 0.009858 0.25166
7 -0.00195 0.249707
8 0.000338 0.250046
9 -5.2E-05 0.249994
10 7.22E-06 0.250001
11 -9.1E-07 0.25
12 1.05E-07 0.25
13 -1.1E-08 0.25
14 1.11E-09 0.25
15 -1E-10 0.25
16 8.89E-12 0.25
17 -7.3E-13 0.25
18 5.59E-14 0.25
19 -4.1E-15 0.25
20 2.82E-16 0.25
0 5 10 15 20 250
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Series2
*According to the graph, the shape becomes back to the first type of graph. Although
the x is negative number, the graph shapes same looking as the graph of T n(a , 1). It might be because of a is half of 1.
a=1/2, x= -2 limn → ∞
T n(12
,−2) = ( 12)−2
= 4
3. Consider the following sequence of terms where x=2 and a=1/2. And calculate the
sum T n of the first n terms of the above sequence for 0≤n≤10.
1, + (2 ln (12))
❑
1 + (2 ln (1
2))
2
2∗1
+ (2 ln (12))
3
3∗2∗1
+ ………..
0≤n≤10
a x n t T
0.5 2 0 1 1
1-
1.38629
-
0.38629
20.96090
6
0.57461
2
3-
0.44403
0.13057
9
4 0.153890.28446
9
5-
0.04267
0.24180
1
60.00985
80.25166
7-
0.00195
0.24970
7
80.00033
8
0.25004
6
9-5.2E-
05
0.24999
4
107.22E-
06
0.25000
1
0 2 4 6 8 10 12
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Series2
0≤n≤20a x n t T
0.5 2 0 1 1
1-1.38629-0.38629
20.9609060.574612
3-0.444030.130579
4 0.153890.284469
5-0.042670.241801
60.009858 0.25166
7-0.001950.249707
80.0003380.250046
9-5.2E-050.249994
107.22E-060.250001
11-9.1E-07 0.25
121.05E-07 0.25
13-1.1E-08 0.25
141.11E-09 0.25
15 -1E-10 0.25
168.89E-12 0.25
17-7.3E-13 0.25
185.59E-14 0.25
19-4.1E-15 0.25
202.82E-16 0.25
0 5 10 15 20 25
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Series2
*According to graph, it becomes exactly same as the graph of T n(2 ,−2) although this
graph’s x value is not negative. This graph is T n(12
,2), but it becomes exact same
looking graph. When n is 2, the sum value becomes negative but when n is 3, the value
goes to positive again. However, when n is 4, the sum value does not go to negative but
it decreases once again. From n becomes 4, the sum value constantly becomes 0.25.
a=1/2, x=2 limn → ∞
T n(12
,2) = 12
2
= 0.25
According to whole observation of investigation, it is proved that the sum value
follows the general statement of ∑n=0
∞ ( lna)n
n!=ax although value of x and a become
rational.
IV. In this part of investigation, how the rational value of x as well as the rational value of a such as 1/3, would affect the value of T n(a, x) would be observed by numerical simulation. In addition, how the irrational value of x as well as the irrational value of x such as π or √ 3 would affect the value of T n(a, x) would be observed by numerical simulation.
1. Consider the following sequence of terms where x=1/3, and a=1/3. And calculate the
sum T n of the first n terms of the above sequence for 0≤n≤10 and 0≤n≤20.
1, + ( 13
ln ( 13))
❑
1 + ( 1
3ln ( 1
3))
2
2∗1
+ ( 13
ln ( 123
))3
3∗2∗1
+ ………..
0≤n≤10
a x n t T
0.3333
33
0.3333
330 1 1
1 -0.36620.6337
96
20.0670
53
0.7008
49
3
-
0.0081
8
0.6926
64
40.0007
49
0.6934
13
5-5.5E-
05
0.6933
58
63.35E-
06
0.6933
61
7-1.8E-
07
0.6933
61
88.02E-
09
0.6933
61
9-3.3E-
10
0.6933
61
10 1.2E-110.6933
61
0 2 4 6 8 10 120
0.2
0.4
0.6
0.8
1
1.2
Series2
0≤n≤20
a x n t T
0.3333
33
0.3333
330 1 1
1 -0.36620.6337
96
20.0670
53
0.7008
49
3
-
0.0081
8
0.6926
64
40.0007
49
0.6934
13
5-5.5E-
05
0.6933
58
63.35E-
06
0.6933
61
7-1.8E-
07
0.6933
61
88.02E-
09
0.6933
61
9-3.3E-
10
0.6933
61
10 1.2E-110.6933
61
11 -4E-130.6933
61
121.21E-
14
0.6933
61
13-3.4E-
16
0.6933
61
148.95E-
18
0.6933
61
15-2.2E-
19
0.6933
61
16 5E-210.6933
61
17-1.1E-
22
0.6933
61
182.19E-
24
0.6933
61
19-4.2E-
26
0.6933
61
20 7.73E- 0.6933
28 61
0 5 10 15 20 250
0.2
0.4
0.6
0.8
1
1.2
Series2
*According to graph, it also shapes like T n(2 ,−2) and T n(12
,2). But the difference is
that sum value never goes to negative although above both of sum value goes to
negative at least once. And from n=5, the sum value constantly get 0.693361.
a=1/3, x=1/3 limn → ∞
T n(13
,13) = 1
3
13 = 0.693361.
2. Consider the following sequence of terms where x=√ 3, and a=π. And calculate the
sum T n of the first n terms of the above sequence for 0≤n≤10.
1, + ((√3 ) ln (π ))❑
1 + ((√3 ) ln (π ))2
2∗1 + ((√3 ) ln (π ))3
3∗2∗1 + ………..
0≤n≤10
a x n t T
π √3 0 1 1
11.9827
3
2.9827
3
21.9656
1
4.9483
4
31.2990
91
6.2474
31
40.6439
37
6.8913
68
50.2553
51
7.1467
19
60.0843
82
7.2311
01
70.0239
01
7.2550
02
80.0059
24
7.2609
25
90.0013
05
7.2622
3
100.0002
59
7.2624
89
0 2 4 6 8 10 120
1
2
3
4
5
6
7
8
Series2
0≤n≤20
a x n t T
3.1415
93
1.7320
510 1 1
11.9827
3
2.9827
3
21.9656
1
4.9483
4
31.2990
91
6.2474
31
40.6439
37
6.8913
68
50.2553
51
7.1467
19
60.0843
82
7.2311
01
70.0239
01
7.2550
02
80.0059
24
7.2609
25
90.0013
05
7.2622
3
100.0002
59
7.2624
89
114.66E-
05
7.2625
36
127.71E-
06
7.2625
43
131.18E-
06
7.2625
45
141.66E-
07
7.2625
45
15 2.2E-087.2625
45
162.73E-
09
7.2625
45
173.18E-
10
7.2625
45
18 3.5E-117.2625
45
193.66E-
12
7.2625
45
203.62E-
13
7.2625
45
0 5 10 15 20 250
1
2
3
4
5
6
7
8
Series2
*According to the graph, it becomes same shape as the graph of T n(2 ,1). When n
becomes 5, the sum value begins to constantly get 7.262545.
According to whole observation of investigation, it is proved that although value of x
and becomes something irrational, sum value can be defined by the general statement of
∑n=0
∞ ( lna)n
n!=a
Conclusion:According to all the observation of data, it can be defined that all of sum value follows
that general statement above. However, the shapes of graphing are not always same.
Some differs from others. From all the data that gathered, sum value of xlna can be
defined that
x a lna xlna
x¿0 a¿1 lna¿0 Positive +
0 2 4 6 8 10 120
10
20
30
40
x¿0 0¿a<¿1 lna<¿0 Negative
0 2 4 6 8 10 12-0.5
0
0.5
1
1.5
Series2
x¿0 a¿1 lna¿0 Negative
0 2 4 6 8 10 12-0.5
0
0.5
1
1.5
Series2
x¿0 0¿a<¿1 lna<¿0 Positive +
0 2 4 6 8 10 120
1
2
3
4
5
Series2
Ultimately, to answer to the aim, whatever the a and x are, infinite sum value follows
the general statement as n increases. So, t n is considered, hence, the value of T n (a, x) as
a n approaches∞,
limn → ∞
T n(a , x)
It can be conclude with
aa¿0
any positive real numbers.
xany real numbers.