The purpose of this paper is to investigate an infinite summation patter where Ln(a) is a constant and the coefficient of x is an increasing factor to Ln(a).
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Introduction
Infinite Summation
Math IA
The purpose of this paper is to investigate an infinite summation patter where Ln(a) is a constant and the coefficient of x is an increasing factor to Ln(a).
Consider the following sequence of terms where x=1 and a=2 under the terms that 0≤n≤10:
tn=
n | t(n) | S(n) |
0.000000 | 1 | 1 |
1.000000 | 0.69314718 | 1.693147 |
2.000000 | 0.24022651 | 1.933373 |
3.000000 | 0.05550411 | 1.988877 |
4.000000 | 0.00961813 | 1.998495 |
5.000000 | 0.00133336 | 1.999829 |
6.000000 | 0.00015404 | 1.999983 |
7.000000 | 1.5253E-05 | 1.999998 |
8.000000 | 1.3215E-06 | 1.999999 |
9.000000 | 1.0178E-07 | 1.999999 |
10.000000 | 7.0549E-09 | 2 |
As n → +∞, Sn → +2
Consider the following sequence of terms where x=1 and a=3:
tn=
n | t(n) | S(n) |
0.000000 | 1.000000 | 1.000000 |
1.000000 | 1.098612 | 2.098612 |
2.000000 | 0.603474 | 2.702087 |
3.000000 | 0.220995 | 2.923082 |
4.000000 | 0.060697 | 2.983779 |
5.000000 | 0.013336 | 2.997115 |
6.000000 | 0.002442 | 2.999557 |
7.000000 | 0.000383 | 2.999940 |
8.000000 | 0.000053 | 2.999993 |
9.000000 | 0.000006 | 2.999999 |
10.000000 | 0.000001 | 3.000000 |
As n → +∞, Sn → +3
There is a horizontal asymptote as n approaches positive infinite (∞). As n approaches positive infinite then Sn will approach positive three. Sn approaches a horizontal asymptote when y=3. There is a y-intercept at (0,1).
As n → +∞, Sn → +4
There is a horizontal asymptote as n approaches positive infinite (∞). As n approaches positive infinite then Sn will approach positive four. Sn approaches a horizontal asymptote when y=4.
Middle
5.666868
3.000000
0.277521
5.944389
4.000000
0.048091
5.992480
5.000000
0.006667
5.999146
6.000000
0.000770
5.999917
7.000000
0.000076
5.999993
8.000000
0.000007
5.999999
9.000000
0.000001
6.000000
As n → +∞, Sn → +6
Let a=2 and calculate various positive values for x:
x | t(n) | S(n) |
0.0 | 1.000000 | 1.000000 |
1.0 | 0.693147 | 1.693147 |
2.0 | 0.480453 | 2.173600 |
3.0 | 0.166512 | 2.340113 |
4.0 | 0.038473 | 2.378585 |
5.0 | 0.006667 | 2.385252 |
In the graph, when x is approaching infinite the Sn values are increasing steadily. When various values are used for x then there is an exponential growth.
Let a=3 then calculate for various positive values of x:
x | t(n) | S(n) |
0.0 | 1.000000 | 1.000000 |
1.0 | 1.098612 | 2.098612 |
2.0 | 1.206949 | 3.305561 |
3.0 | 0.662984 | 3.968546 |
4.0 | 0.242788 | 4.211333 |
5.0 | 0.066682 | 4.278016 |
In the graph, when x is approaching infinite the Sn values are increasing steadily. When various values are used for x then there is an exponential growth.
Evidence:
tn=
n | t(n) | S(n) |
0.000000 | 1.000000 | 1.000000 |
1.000000 | 2.772589 | 3.772589 |
2.000000 | 0.960906 | 4.733495 |
3.000000 | 0.222016 | 4.955511 |
4.000000 | 0.038473 | 4.993984 |
5.000000 | 0.005333 | 4.999317 |
6.000000 | 0.000616 | 4.999933 |
7.000000 | 0.000061 | 4.999994 |
8.000000 | 0.000005 | 5.000000 |
9.000000 | 0.000000 | 5.000000 |
As n → +∞, Sn → +6
Sn approaches a horizontal asymptote when y=6. There is a y-intercept at (0,1).
tn =
n | t(n) | S(n) |
0.000000 | 1.000000 | 1.000000 |
1.000000 | 4.158883 | 5.158883 |
2.000000 | 1.441359 | 6.600242 |
3.000000 | 0.333025 | 6.933267 |
4.000000 | 0.057709 | 6.990976 |
5.000000 | 0.008000 | 6.998976 |
6.000000 | 0.000924 | 6.999900 |
7.000000 | 0.000092 | 6.999991 |
8.000000 | 0.000008 | 6.999999 |
9.000000 | 0.000001 | 7.000000 |
As n → +∞, Sn → +7
There is a horizontal asymptote as n approaches positive infinite (∞). As n approaches positive infinite then Sn will approach positive seven. Sn approaches a horizontal asymptote when y=7. There is a y-intercept at (0,1).
tn =
n | t(n) | S(n) |
0.000000 | 1.000000 | 1.000000 |
1.000000 | 4.852030 | 5.852030 |
2.000000 | 1.681586 | 7.533616 |
3.000000 | 0.388529 | 7.922145 |
4.000000 | 0.067327 | 7.989471 |
5.000000 | 0.009333 | 7.998805 |
6.000000 | 0.001078 | 7.999883 |
7.000000 | 0.000107 | 7.999990 |
8.000000 | 0.000009 | 7.999999 |
9.000000 | 0.000001 | 8.000000 |
As n → +∞, Sn → +8
There is a horizontal asymptote as n approaches positive infinite (∞). As n approaches positive infinite then Sn will approach positive eight. Sn approaches a horizontal asymptote when y=8. There is a y-intercept at (0,1).
tn =
n | t(n) | S(n) |
0.000000 | 1.000000 | 1.000000 |
1.000000 | 5.545177 | 6.545177 |
2.000000 | 1.921812 | 8.466990 |
3.000000 | 0.444033 | 8.911022 |
4.000000 | 0.076945 | 8.987967 |
5.000000 | 0.010667 | 8.998634 |
6.000000 | 0.001232 | 8.999867 |
7.000000 | 0.000122 | 8.999989 |
8.000000 | 0.000011 | 8.999999 |
9.000000 | 0.000001 | 9.000000 |
As n → +∞, Sn → + 9
There is a horizontal asymptote as n approaches positive infinite (∞). As n approaches positive infinite then Sn will approach positive nine. Sn approaches a horizontal asymptote when y=9. There is a y-intercept at (0,1).
tn =
n | t(n) | S(n) |
0.000000 | 1.000000 | 1.000000 |
1.000000 | 6.238325 | 7.238325 |
2.000000 | 2.162039 | 9.400363 |
3.000000 | 0.499537 | 9.899900 |
4.000000 | 0.086563 | 9.986463 |
5.000000 | 0.012000 | 9.998464 |
6.000000 | 0.001386 | 9.999850 |
7.000000 | 0.000137 | 9.999987 |
8.000000 | 0.000012 | 9.999999 |
9.000000 | 0.000001 | 10.000000 |
10.000000 | 0.000000 | 10.000000 |
Conclusion
tn =
n | t(n) | S(n) |
0.000000 | 1.000000 | 1.000000 |
1.000000 | 5.493061 | 6.493061 |
2.000000 | 3.017372 | 9.510434 |
3.000000 | 1.104974 | 10.615408 |
4.000000 | 0.303485 | 10.918893 |
5.000000 | 0.066682 | 10.985575 |
6.000000 | 0.012210 | 10.997785 |
7.000000 | 0.001916 | 10.999701 |
8.000000 | 0.000263 | 10.999964 |
9.000000 | 0.000032 | 10.999996 |
As n → +∞, Sn → + 11
There is a horizontal asymptote as n approaches positive infinite (∞).
tn=
n | t(n) | S(n) |
0.0 | 1.000000 | 1.000000 |
1.0 | 7.690286 | 8.690286 |
2.0 | 4.224321 | 12.914607 |
3.0 | 1.546964 | 14.461571 |
4.0 | 0.424878 | 14.886450 |
5.0 | 0.093355 | 14.979805 |
6.0 | 0.017094 | 14.996898 |
7.0 | 0.002683 | 14.999581 |
8.0 | 0.000368 | 14.999950 |
9.0 | 0.000045 | 14.999995 |
As n → +∞, Sn → + 15
There is a horizontal asymptote as n approaches positive infinite (∞). As n approaches positive infinite then Sn will approach positive fifteen. Sn approaches a horizontal asymptote when y=15. There is a y-intercept at (0,1).
tn=
n | t(n) | S(n) |
0.0 | 1.000000 | 1.000000 |
1.0 | 8.788898 | 9.788898 |
2.0 | 4.827796 | 14.616694 |
3.0 | 1.767959 | 16.384653 |
4.0 | 0.485575 | 16.870228 |
5.0 | 0.106692 | 16.976920 |
6.0 | 0.019535 | 16.996455 |
7.0 | 0.003066 | 16.999521 |
8.0 | 0.000421 | 16.999942 |
9.0 | 0.000051 | 16.999994 |
As n → +∞, Sn → + 17
There is a horizontal asymptote as n approaches positive infinite (∞). Sn approaches a horizontal asymptote when y=17. There is a y-intercept at (0,1).
Checks with various numbers:
tn =
n | t(n) | S(n) |
0.0 | 1.000000 | 1.000000 |
1.0 | 48.283137 | 49.283137 |
2.0 | 38.854356 | 88.137493 |
3.0 | 20.844558 | 108.982051 |
4.0 | 8.387005 | 117.369057 |
5.0 | 2.699673 | 120.068729 |
6.0 | 0.724159 | 120.792889 |
7.0 | 0.166498 | 120.959387 |
8.0 | 0.033496 | 120.992883 |
9.0 | 0.005990 | 120.998873 |
10.000000 | 0.000964 | 120.999837 |
tn =
n | t(n) | S(n) |
0.0 | 1.000000 | 1.000000 |
1.0 | 13.961881 | 14.961881 |
2.0 | 32.489020 | 47.450901 |
3.0 | 50.400871 | 97.851772 |
4.0 | 58.640914 | 156.492686 |
5.0 | 54.582497 | 211.075184 |
6.0 | 42.337463 | 253.412647 |
7.0 | 28.148125 | 281.560772 |
8.0 | 16.375032 | 297.935804 |
9.0 | 8.467639 | 306.403443 |
10.000000 | 3.940806 | 310.344249 |
n | t(n) | S(n) |
0.0 | 1.000000 | 1.000000 |
1.0 | 179.175947 | 180.175947 |
2.0 | 160.520100 | 340.696047 |
3.0 | 95.871136 | 436.567183 |
4.0 | 42.944504 | 479.511687 |
5.0 | 15.389244 | 494.900931 |
6.0 | 4.595637 | 499.496569 |
7.0 | 1.176325 | 500.672894 |
8.0 | 0.263461 | 500.936356 |
9.0 | 0.052451 | 500.988807 |
10.000000 | 0.009398 | 500.998205 |
tn =
Conclusion:
The limitation to this general statement is that the scope of the evidence is limited to only a few combinations of x and a .
The continuous observation of various graphs with different values used in place of a and x provide evidence for the general statement.
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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