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 , we write the infinite series as
When adding up only the first n terms of a sequence, we refer to the partial sum:
So, there are two sequences associated with any series :
•

, the sequence of its terms

•

, the sequence of its partial sums

A series is said to converge if the sequence of its partial sums, , converge. The finite limit of 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 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 dropdown 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.




Plot the sequence for





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