# 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)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)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)Therefore:

(4)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.