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# Series

A series is a group of similar things that are all related to the same topic.

In mathematics, a series is the adding of a list of (usually never-ending) mathematical objects (such as numbers). It is sometimes written as $\textstyle \sum_{n=i}^k a_n$,[1] which is another way of writing $a_i + \cdots + a_k$.

For example, the series $\textstyle \sum_{n=0}^{\infty} \frac{1}{2^n}$[2] corresponds to the following sum:

$1 + \frac12 + \frac14 + \frac18 + \frac1{16} + \frac1{32} + \frac1{64} + \frac1{128} + \ldots$

Here, the dots mean that the adding does not have a last term, but goes on to infinity.

If the result of the addition gets closer and closer to a certain limit value, then this is the sum of the series. For example, the first few terms of the above series are:

$1 + \frac12 = 1 \frac12$

$1 + \frac12 + \frac14 = 1 \frac34$

$1 + \frac12 + \frac14 + \frac18 = 1 \frac78$

$1 + \frac12 + \frac14 + \frac18 + \frac1{16} = 1 \frac{15}{16}$

$1 + \frac12 + \frac14 + \frac18 + \frac1{16} + \frac1{32} = 1 \frac{31}{32}$

$1 + \frac12 + \frac14 + \frac18 + \frac1{16} + \frac1{32} + \frac1{64} = 1 \frac{63}{64}$

$1 + \frac12 + \frac14 + \frac18 + \frac1{16} + \frac1{32} + \frac1{64} + \frac1{128} = 1 \frac{127}{128}$

From these, we can see that this series will have 2 as its sum.

However, not all series have a sum. For example. a series can go to positive or negative infinity, or just go up and down without settling on any particular value. In which case, the series is said to diverge.[3] The harmonic series is an example of a series which diverges.

## References

1. Weisstein, Eric W.. "Series" (in en).