Series - Maple Help

Series

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

When adding up only the first n terms of a sequence, we refer to the ${\mathbit{n}}^{\mathbit{th}}$ partial sum:

So, there are two sequences associated with any series $\sum {a}_{n}$ :

 • $\left\{{a}_{n}\right\}$, the sequence of its terms
 • $\left\{{s}_{n}\right\}$, the sequence of its partial sums

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:

.

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.

Choose a closed formula for a sequence from the drop-down menu below, or type your own formula in the text box and click "Enter" to see a plot of the first N partial sums. Use the slider to adjust how many points are plotted and select the check box to find out if this sequence converges or diverges.

 Enter Functionn1/n1 + 3*(n - 1)1/(2^n)n/(3^n)n^10/(2^n)(2*n + 1)^21/(n-1)!2^(n - 1)cos(n)/nsin(1/n)sin(1/(n^2)) + cos(1/n)(-1)^n/n(-1)^n/sqrt(n)-1/(-2)^nn*(-1)^n

Plot the sequence for

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