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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 an, we write the infinite series as n=1an = a1 + a2 + a3 + ... + an1 + an + an+1 + ...

When adding up only the first n terms of a sequence, we refer to the nth partial sum: sn = i=1nai = a1 + a2 + ... + an

So, there are two sequences associated with any series an :

• 

 an, the sequence of its terms

• 

 sn, the sequence of its partial sums

 

A series is said to converge if the sequence of its partial sums, sn, converge. The finite limit of sn as n approaches infinity is then called the sum of the series:

S =limnsn =n=1an.

This means that by adding sufficiently many terms of the series, we can get very close to the value of S. If sn 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.

an = 


 

Plot the sequence for n = 1 ..

 

 

 

 

 

 

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MathApps/Calculus

 


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