Introduction

Welcome to this guide on defining Convergent and Divergent Infinite Series. This article will guide you through the following:

Basic Terms

To start Unit 10, we are going to introduce to some terms about this unit:

Convergence and Divergence of Infinite Series

With this we can define the convergence and divergence of infinite series.

Let $S_n$ be the partial sum of a sequence for the first $n$ terms, if 

$\underset{n\to \infty }{\lim }{S}_n=L$

Then we say this infinite series converges to $L$. If the limit does not exist, we say this infinite series diverges. We can also understand the convergence of a series as a sequence with a finite sum, whereas divergent series are sequences with a infinite sum.

Denoting Series

In general, we use Greek letter $\Sigma$ (pronounced as "sigma") to represent sum, it is used like this:

$\sum _{n=\text{Starting Index}}^{\text{Final Index}}n\text{th term formula}$

The $n$ represents the summation index, where the starting and final index dictate where the series should start and where it should end, respectively. Meanwhile, the $n\text{th term formula}$ dictates what each individual term of the series should be. 

In mathematics, oftentimes, the starting index can either be designated as $n=0$ or $n=1$. EXTREME care must be made in distinguishing between the two cases, which will be discussed more in depth later. Zero-based indexing often has practical applications in computer science and applied mathematics, whereas one-based indexing is used in other applications.

Example 1

$\sum _{n=1}^{\infty }\frac{1}{n^2}=1+\frac{1}{2^2}+\frac{1}{3^2}+...$

The above is an example of the sum of all the reciprocal of natural numbers squared. It should thus make logical sense that as Unit 10 deals with infinite series, the final index for them would naturally be $\infty$.

(This series is also called the Basel Problem, which can be proven to converge via various methods. This problem will be discussed more in depth later.)

Example 2

$\sum _{n=1}^5{n}^2=1^2+2^2+3^2+4^2+5^2$

means adding $1^2$ to $5^2$ (a finite series), while

$\sum _{n=1}^{\infty }{n}^2=1^2+2^2+3^2+4^2+5^2+...$

means adding $1^2$ all the way to infinity.

Practice