Main Concept

In mathematics, a series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, while infinite sequences and series continue on indefinitely.

Given an infinite sequence $\left\{a_n\right\}$, we write the infinite series as $\sum _{n\=1}^{\infty }{a}_n \= a_1 \+ a_2 \+ a_3 \+ ..\. \+ a_{n-1} \+ a_n \+ a_{n\+1} \+ ..\.$

When adding up only the first n terms of a sequence, we refer to the ${\mathit{n}}^{\mathit{th}}$ partial sum: $s_n \= \sum _{i\=1}^n{a}_i \= a_1 \+ a_2 \+ ..\. \+ a_n$

So, there are two sequences associated with any series $\sum a_n$ :

A series is said to converge if the sequence of its partial sums, $\left\{s_n\right\}$, converge. The finite limit of $s_n$ as n approaches infinity is then called the sum of the series:

$S \=\underset{n\to \infty }{lim}{s}_n \=\sum _{n\=1}^{\infty }{a}_n$.

This means that by adding sufficiently many terms of the series, we can get very close to the value of S. If $\left\{s_n\right\}$ diverges, then the series diverges as well.

Finding S is often very difficult, and so the main focus when working with series is often just testing to figure out whether the series converges or diverges.

Plot the sequence for $n \= 1 ..$

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