Infinite Series

An infinite series is the summation of all the terms in an infinite sequence. We use the sigma notation for summations.

$$\displaystyle \sum^{\infty}_{n=0}a_n=a_0+a_1+a_2+a_3+\dots$$

Again we are concerened about whether the series adds up to a finite value. If it does then we say the series converges. Otherwise we say the series diverges.

The first step in deciding whether a series converges or diverges is to use the ``$n$th term test for divergence.'' If the limit of the terms in the sequence is not zero then the series diverges; i.e., $\lim_{n\to\infty}a_n\ne 0\Longrightarrow$ divergence. This is only a test for divergence. If $\lim_{n\to\infty}a_n=0$ the test fails and we have to find some other technique.

To determine whether a series converges we need to look at the $n$th partial sums denoted $S_n$. If we can find a formula for $S_n$ the we can decide immediately what the first $n$ terms of the series add up to. Then we get the actual sum of the infinite series as

$$\displaystyle \sum^{\infty}_{n=1}a_n=\displaystyle \lim_{n\to\infty}S_n$$