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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 neverending) mathematical objects (such as numbers). It is sometimes written as [math]\textstyle \sum_{n=i}^k a_n[/math],^{[1]} which is another way of writing [math]a_i + \cdots + a_k[/math].
For example, the series [math]\textstyle \sum_{n=0}^{\infty} \frac{1}{2^n}[/math]^{[2]} corresponds to the following sum:
 [math]1 + \frac12 + \frac14 + \frac18 + \frac1{16} + \frac1{32} + \frac1{64} + \frac1{128} + \ldots [/math]
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:
 [math]1 + \frac12 = 1 \frac12[/math]
 [math]1 + \frac12 + \frac14 = 1 \frac34[/math]
 [math]1 + \frac12 + \frac14 + \frac18 = 1 \frac78[/math]
 [math]1 + \frac12 + \frac14 + \frac18 + \frac1{16} = 1 \frac{15}{16}[/math]
 [math]1 + \frac12 + \frac14 + \frac18 + \frac1{16} + \frac1{32} = 1 \frac{31}{32}[/math]
 [math]1 + \frac12 + \frac14 + \frac18 + \frac1{16} + \frac1{32} + \frac1{64} = 1 \frac{63}{64}[/math]
 [math]1 + \frac12 + \frac14 + \frac18 + \frac1{16} + \frac1{32} + \frac1{64} + \frac1{128} = 1 \frac{127}{128}[/math]
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.
Related page
References
 ↑ "List of Calculus and Analysis Symbols" (in enUS). 20200511. https://mathvault.ca/hub/highermath/mathsymbols/calculusanalysissymbols/.
 ↑ Weisstein, Eric W.. "Series" (in en). https://mathworld.wolfram.com/Series.html.
 ↑ "Infinite Series". https://www.mathsisfun.com/algebra/infiniteseries.html.
