1. Fuyuki Watanabe
IB Math SL
Ms. Bessette
Period 5
March 27, 2010
IB Math Internal Assessment:
Infinite Summation
Introduction:
The aim of this internal assessment is to investigate the sum of infinite sequences. The
infinite sequence that is being investigated is . The notation in the denominator is
called the factorial or Factorial is the product of all positive integers less than or equal to n.
Data:
The sum of the first n terms of the sequence above for when x = 1 and a = 2
is…
n Sum
Sum of the Sequence (x = 1 a = 2)
0 1.000000 10
9
1 1.693147
8
2 1.933374 7
3 1.988878 6
Value of Sum
5
4 1.998496 4
5 1.999829 3
6 1.999983 2
1
7 1.999999 0
8 2.000000 0 2 4 6 8 10 12
9 2.000000 Value of n
10 2.000000
Graph 1 – shows the sum of the infinite sequence when x = 1 and a =
2. When n = 0, the sum is 1. As n increases, the sum increases rapidly
until it starts to level off when n = 2. This may suggest that, without
rounding off, the value of the sum will get closer to 2 but never reach
2 as n approaches ∞.

2. Now when the sequence of same terms where this time x = 1 and a = 3, the sum of the first
n terms of the sequence is…
n Sum Sum of the Sequence (x = 1, a = 3)
0 1.000000 10
1 2.098612 9
8
2 2.702087
7
Value of Sum
3 2.923082 6
4 2.983779 5
4
5 2.997115 3
6 2.999557 2
1
7 2.99994 0
8 2.999993 0 2 4 6 8 10 12
9 2.999999 Value of n
10 3.000000
Graph 2 – shows the sum of the infinite sequence when x = 1 and a =
3. When n = 0, the sum is 1. As n increases, the sum increases rapidly
until it starts to level off when n = 3. This may suggest that, without
rounding off, the value of the sum will get closer to 3 but never reach
3 as n approaches ∞.
Now when the sequence of the same terms where this time x =1 and a = 4, 5, or 8, the sum
of the first n terms of the sequence is…
n Sum (a =4) Sum (a =5) Sum (a =8)
0 1 1 1
1 2.386294 2.609438 3.079442
2 3.347200 3.904583 5.241480
3 3.791233 4.599402 6.740091
4 3.945123 4.878969 7.519159
5 3.987791 4.968958 7.843165
6 3.997649 4.993096 7.955457
7 3.999601 4.998646 7.988814
8 3.999940 4.999763 7.997485
9 3.999992 4.999962 7.999488
10 3.999999 4.999995 7.999905
*x = 1 for all values

3. Sum of the Sequence
10
9
8
a=4
7
Value of Sum
6
5
a=5
4
3
2 a=8
1
0
0 2 4 6 8 10 12
Value of n
Graph 3 – shows the sum of the infinite sequence when x = 1 and a = 4, 5, or 8. For all the value of
a, when n = 0, the sum is 1. For each value of a, as n increases, the sum increases rapidly until it
starts to level off when the sum equals whatever the value of a. (ex. when a = 4, the sum levels off
at 4). This may suggest that, without rounding off, the value of the sum will get closer to whatever
the value of a but never reach its value as n approaches ∞.
From observing the effect of altering the value of a while x remains as 1, it can be generalized that
the infinite sum of this general sequence is the value of a. The data shows that when x = 1 and a = 1,
2, 3, 4, 5, or 8, without rounding up, the sum the sequence for the first ten terms was close to 1, 2, 3,
4, 5, or 8 respectively. This may suggest that a horizontal asymptote may be present at whatever the
value of a when x = 1.
The sum of n terms for the sequence < a
(Later mathematical processing supports that this is only true when x = 1 because )

4. Now the research will investigate the effect of altering the value of x when a is constant.
The sum of the first 9 terms for the sequence , when x = 3, 4, or 5 and a = 2 is…
n Sum (x = 3) Sum (x = 4) Sum (x = 5)
0 1 1 1
1 3.079442 3.772589 4.465736
2 5.241480 7.616213 10.47140
3 6.740091 11.16848 17.40941
4 7.519159 13.63072 23.42074
5 7.843165 14.99607 27.58748
6 7.955457 15.62700 29.99428
7 7.988814 15.87690 31.18590
8 7.997485 15.96351 31.70213
9 7.999488 15.99019 31.90092
*a = 2 for all the values
Sum of the Sequence
36
32
28
24 x=3
Value of Sum
20
16 x=4
12
x=5
8
4
0
0 2 4 6 8 10
Value of n
Graph 4 – shows the sum of the infinite sequence when x = 3, 4, or 5 and a = 2. For all the value of
x, when n = 0, the sum is 1. For each value of x, as n increases, the sum increases rapidly until it
starts to level off when the sum equals whatever the value of (ex. when x = 4, the sum levels off
at ).

5. Now when x = 2, 3, or 4 and a = 3, the sum of the first 9 terms for the sequence
is…
n Sum (x = 2) Sum (x = 3) Sum (x = 4)
0 1 1 1
1 3.197225 4.295837 5.394449
2 5.611122 9.727107 15.05004
3 7.379081 15.69397 29.19371
4 8.350232 20.61042 44.73212
5 8.776999 23.85118 58.38867
6 8.933283 25.63135 68.39084
7 8.982339 26.46952 74.66998
8 8.995812 26.81482 78.11916
9 8.999101 26.94128 79.80329
*a = 3 for all the values
Sum of the Sequence
81
72
63
x=2
54
Value of sum
45
x=3
36
27
x=4
18
9
0
0 2 4 6 8 10
Value of n
Graph 5 – shows the sum of the infinite sequence when x = 2, 3, or 4 and a = 3. For all the value of
x, when n = 0, the sum is 1. For each value of x, as n increases, the sum increases rapidly until it
starts to level off when x equals whatever the value of (ex. when x = 4, the sum levels off
at ).

6. From observing the effect of altering the value of x while a remains constant, it can be generalized
that the infinite sum of this general sequence is the value of . The data shows that when x = 3,4, or
5 and a = 2, without rounding up, the sum the sequence for the first nine terms was close to 8, 16,
and 32 respectively. Also, when x = 2, 3, or 4 and a = 3, without rounding up, the sum of the
sequence for the first nice terms was close to 9, 27, and 81 respectively. This may suggest that a
horizontal asymptote may be present at whatever the value of .
The sum of n terms for the sequence <

7. To test the validity of this finding, other values of a and x were tested…
N x=2;a=6 x=3;a=5 x=4;a=4
0 1 1 1
1 4.583519 5.828314 6.545177
2 11.00432 17.48462 21.91967
3 18.67401 36.24472 50.33778
4 25.54513 58.88964 89.73363
5 30.46969 80.75699 133.425
6 33.4109 98.35406 173.8045
7 34.9166 110.4918 205.7918
8 35.59106 117.8174 227.9637
9 35.85961 121.7474 241.6245
10 35.95584 123.645 249.1996
11 35.98719 124.4779 253.0183
12 35.99656 124.813 254.7829
13 35.99914 124.9375 255.5356
14 35.9998 124.9804 255.8338
15 35.99996 124.9942 255.944
Sum of the Sequence
300
250
Value of Sum
200
150 x=2;a=6
100 x=3;a=5
50 x=4;a=4
0
0 2 4 6 8 10 12 14 16
Value of n
Graph 6 – shows the sum of the infinite sequence when x = 2 & a = 6, x= 3 & a = 5, and x = 4 & a =
4. For all the value of x and a, when n = 0, the sum is 1. This is because any value to the power of 0
= 1. As expected, for each set, the sum increases rapidly as n increases until it starts to level off
when the sum equals whatever the value of . When x = 2 and a = 6, it starts to level off at or 36.
When x = 3 and a = 5, it starts to level off at or 125. When x= 2 and a = 6, it starts to level off at or 256.

8. The scope of the general statement is true to some extent; however there are some limitations that
may affect the result. Because the numbers were rounded to the sixth decimal place, the graph can
be misleading as it does not represent the true value. Also, it is impossible to calculate the infinite
number of sequence and any natural log of a negative number is an error.
The general statement was produced through several mathematical processing of data. First, the
effect of altering the value a while x = 1 was investigated. The result supported that there is a
horizontal asymptote of y = a. Next, the effect of altering the value of x while a remains constant
was investigated. The result supported that there is a horizontal asymptote of y = . This supports
that the effect of altering the value a while = 1 was actually producing a horizontal asymptote of y
= A few more samples were tested to validate this trend.
Academic Honesty
“I, the undersigned, hereby, declare that the following assignment is all my own work and that I
worked independently on it.”
“In this assignment, I used Microsoft Excel 2007 to draw my graphs.”