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Series



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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 [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

  1. "List of Calculus and Analysis Symbols" (in en-US). 2020-05-11. https://mathvault.ca/hub/higher-math/math-symbols/calculus-analysis-symbols/. 
  2. Weisstein, Eric W.. "Series" (in en). https://mathworld.wolfram.com/Series.html. 
  3. "Infinite Series". https://www.mathsisfun.com/algebra/infinite-series.html.