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

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

[math]\displaystyle{ 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]\displaystyle{ 1 + \frac12 = 1 \frac12 }[/math]

[math]\displaystyle{ 1 + \frac12 + \frac14 = 1 \frac34 }[/math]

[math]\displaystyle{ 1 + \frac12 + \frac14 + \frac18 = 1 \frac78 }[/math]

[math]\displaystyle{ 1 + \frac12 + \frac14 + \frac18 + \frac1{16} = 1 \frac{15}{16} }[/math]

[math]\displaystyle{ 1 + \frac12 + \frac14 + \frac18 + \frac1{16} + \frac1{32} = 1 \frac{31}{32} }[/math]

[math]\displaystyle{ 1 + \frac12 + \frac14 + \frac18 + \frac1{16} + \frac1{32} + \frac1{64} = 1 \frac{63}{64} }[/math]

[math]\displaystyle{ 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". Math Vault. 2020-05-11. Retrieved 2020-08-30.
  2. Weisstein, Eric W. "Series". mathworld.wolfram.com. Retrieved 2020-08-30.
  3. "Infinite Series". www.mathsisfun.com. Retrieved 2020-08-30.