# Infinite Summation - IB Math SL Mathematics Portfolio 2011

by cartoonlover (student)

Anglo-Chinese School (Independent)

Infinite Summation

“I undersigned, hereby declare that the following course work is all my own work and that I worked independently on it”

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## Introduction

A summation is the operation of adding a sequence of numbers, with the result being their sum total. An infinite series is essentially a series with an infinite number of terms. Due to the indeterminable nature of all the terms, mathematical analysis is required to fully understand the series and its properties. However, that is not to say that infinite series are not used often. In fact, they are regularly used in the fields of physics and computer science. In this portfolio, we will be investigating the sum of infinite sequences tn where

t0=1, t1=(xlna)1, t2=(xlna)22!, t3=(xlna)33!,…,tn=(xlna)nn!

By observing various trends within the series, I will come up with some general statements and will test the validity of said statements with the aid of some examples. Then, I will try to link the patterns discovered to some pre-established mathematical concepts, such as the Maclaurin series.

## Part I

Firstly, let us look at the sequence of terms when x=1 and a=2, as the value of n increases.

When n=10, we can see that S    n is equal to 2.000000. The graph shows us that as n tends towards ∞, S    n increases and tends towards the value of 2.000000.

Now, let us look at the same equation when x=1 and a=3 as the value of n increases.

Again, as n tends towards ∞, S    n increases and tends towards the value of 3.000000.

We have tried the sequence in relation to positive integers. Now, let us investigate the sequence in relation to a few other values of a as x=1. We ...