Series

Infinite sum

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 ${\displaystyle \textstyle \sum _{n=i}^{k}a_{n}}$,[1] which is another way of writing ${\displaystyle a_{i}+\cdots +a_{k}}$.

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

${\displaystyle 1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+{\frac {1}{32}}+{\frac {1}{64}}+{\frac {1}{128}}+\ldots }$

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:

${\displaystyle 1+{\frac {1}{2}}=1{\frac {1}{2}}}$

${\displaystyle 1+{\frac {1}{2}}+{\frac {1}{4}}=1{\frac {3}{4}}}$

${\displaystyle 1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}=1{\frac {7}{8}}}$

${\displaystyle 1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}=1{\frac {15}{16}}}$

${\displaystyle 1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+{\frac {1}{32}}=1{\frac {31}{32}}}$

${\displaystyle 1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+{\frac {1}{32}}+{\frac {1}{64}}=1{\frac {63}{64}}}$

${\displaystyle 1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+{\frac {1}{32}}+{\frac {1}{64}}+{\frac {1}{128}}=1{\frac {127}{128}}}$

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. "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.