Convergence and Divergence of Infinite Series

# Convergence and Divergence of Infinite Series

Recall from the Infinite Series of Real and Complex Numbers page that if $(a_n)_{n=1}^{\infty}$ is an infinite sequence of real/complex numbers (known as the sequence of terms) then the corresponding series is the infinite sum of the terms in this sequence:

(1)
\begin{align} \quad \sum_{n=1}^{\infty} a_n = a_1 + a_2 + ... \end{align}

Furthermore, we defined the corresponding sequence of partial sums $(s_n)_{n=1}^{\infty}$ to be defined with the general $n^{\mathrm{th}}$ term given by:

(2)
\begin{align} \quad s_n = \sum_{k=1}^{n} a_k = a_1 + a_2 + ... + a_n \end{align}

We will now discuss the important concept of convergence/divergence of an infinite series.

 Definition: A series $\displaystyle{\sum_{n=1}^{\infty} a_n}$ is said to Converge to the sum $s$ if the sequence of partial sums $(s_n)_{n=1}^{\infty}$ converges to $s$, i.e., $\displaystyle{\lim_{n \to \infty} s_n = s}$. A series is said to Diverge if it does not converge to any sum.

Note that in general, determining whether a series converges or diverges can be rather difficult. The general term of the sequence of partial sums is $s_n = \sum_{k=1}^{n} a_k$ may not be too easy to work with in terms of limits. Sometimes we can instead rewrite the general term out of sigma notation.

For example, consider the series $\sum_{k=1}^{\infty} \left ( \frac{1}{k+1} - \frac{1}{k} \right )$. Then the general term for the sequence of partial sums is:

(3)
\begin{align} \quad s_n &= \sum_{k=1}^{n} \left ( \frac{1}{k+1} - \frac{1}{k} \right ) \\ \quad s_n &= \left ( \frac{1}{2} - 1 \right ) + \left ( \frac{1}{3} - \frac{1}{2} \right ) + ... + \left ( \frac{1}{n+1} - \frac{1}{n} \right ) \\ \quad s_n &= \frac{1}{n+1} - 1 \end{align}

Therefore:

(4)
\begin{align} \quad \lim_{n \to \infty} s_n = \lim_{n \to \infty} \left ( \frac{1}{n + 1} - 1 \right ) = -1 \end{align}

So the series $\displaystyle{\sum_{k=1}^{\infty} s_k = - 1}$. The series above is actually a special type of series known as a telescoping series which we will look more into later on.