# Definition:Series/Number Field

## Definition

Let $S$ be one of the standard number fields $\R$, or $\C$.

Let $\sequence {a_n}$ be a sequence in $S$.

The **series** is what results when $\sequence {a_n}$ is summed to infinity:

- $\displaystyle \sum_{n \mathop = 1}^\infty a_n = a_1 + a_2 + a_3 + \cdots$

### Real Series

A **real series** $S_n$ is the limit to infinity of the sequence of partial sums of a real sequence $\sequence {a_n}$:

\(\displaystyle S_n\) | \(=\) | \(\displaystyle \lim_{N \mathop \to \infty} \sum_{n \mathop = 1}^N a_n\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \sum_{n \mathop = 1}^\infty a_n\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle a_1 + a_2 + a_3 + \cdots\) |

### Complex Series

A **complex series** $S_n$ is the limit to infinity of the sequence of partial sums of a complex sequence $\sequence {a_n}$:

\(\displaystyle S_n\) | \(=\) | \(\displaystyle \lim_{N \mathop \to \infty} \sum_{n \mathop = 1}^N a_n\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \sum_{n \mathop = 1}^\infty a_n\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle a_1 + a_2 + a_3 + \cdots\) |

## Historical Note

Much of the original work on series of real and complex numbers was done by Leonhard Paul Euler.

The main bulk of the work to placed the concept on a rigorous footing was done by Carl Friedrich Gauss, Niels Henrik Abel and Augustin Louis Cauchy.

## Sources

- 1992: Larry C. Andrews:
*Special Functions of Mathematics for Engineers*... (previous) ... (next): $\S 1.2$: Infinite Series of Constants