John Frontczak

Period 8 IB SL Math

1/04/10

Infinite Summation:

The aim of this task is to investigate the sum of infinite sequences, of the mathematic statement:

Investigation:

Calculating where x=1, a=2, and 0≤𝑛≤10.

As n get larger, get exponentially smaller.

Graph 1: Showing the relationship between vs n.

The , as n increases, seems to reach an isotope that is close to (if not)

Table: where (T), & (t), when a=2 &

x=2.

Series 1

Series 2

Series 3 Series 5

Series 6

Series 7

Series 4

Table: where (T), & (t), when a=2 &

x=4.

Table: where (T), & (t), when a=2 &

x=6.

Table: where (T), & (t), when a=4 &

x=2.

Table: where (T), & (t), when a=5 &

x=2.

Table: where (T), & (t), when a=6 & x=2

Graph 2: is plotted against x, where .

appears to increase exponentially as x increases.

General Statement:

, as approaches , it reaches an isotope at .

Discussion:

In each of the sums calculated, each approached and got close to the isotope predicted, while never actually getting. However, the sum of the terms never become greater than the

predicted isotope, even after the 20th term. This lead me to conclude that , as

approaches , reaches an isotope at . After testing this seemed to be true for decimals,

Series 1 Series 4 Series 2 Series 3

positive integers and negative integers between -10 and 10, thus leading me to believe the general statement I had arrived at.

While the general statement seems to be true, really the sum never reaches a to the power of x as it is an isotope, and as such does not reach . Also, it was only checked whether this was true for number between -10 and 10, and it is quite possible that the statement stops being, even remotely true, past these integers.