Series: Infinite sum

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

For example, the series ${\displaystyle \textstyle \sum _{n=0}^{\infty }{\frac {1}{2^{n}}}}$ 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. The harmonic series is an example of a series which diverges.

• List of series
• Sequence
• Series acceleration
• Taylor series