# 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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