Definition:Convergent Series/Number Field
Definition
Let $S$ be one of the standard number fields $\Q, \R, \C$.
Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be a series in $S$.
Let $\sequence {s_N}$ be the sequence of partial sums of $\ds \sum_{n \mathop = 1}^\infty a_n$.
It follows that $\sequence {s_N}$ can be treated as a sequence in the metric space $S$.
If $s_N \to s$ as $N \to \infty$, the series converges to the sum $s$, and one writes $\ds \sum_{n \mathop = 1}^\infty a_n = s$.
A series is said to be convergent if and only if it converges to some $s$.
Examples of Convergent Complex Series
Example: $\dfrac {\paren {-1}^n + i \cos n \theta} {n^2}$
The series $\ds \sum_{n \mathop = 1}^\infty a_n$, where:
- $a_n = \dfrac {\paren {-1}^n + i \cos n \theta} {n^2}$
is convergent.
Example: $\dfrac 1 {n^2 - i n}$
The series $\ds \sum_{n \mathop = 1}^\infty a_n$, where:
- $a_n = \dfrac 1 {n^2 - i n}$
is convergent.
Example: $\dfrac {e^{i n} } {n^2}$
The series $\ds \sum_{n \mathop = 1}^\infty a_n$, where:
- $a_n = \dfrac {e^{i n} } {n^2}$
is convergent.
Example: $\paren {\dfrac {2 + 3 i} {4 + i} }^n$
The series $\ds \sum_{n \mathop = 1}^\infty a_n$, where:
- $a_n = \paren {\dfrac {2 + 3 i} {4 + i} }^n$
is convergent.
Also see
- Results about convergent series can be found here.
Historical Note
Many of the basic tests for convergence of series have Augustin Louis Cauchy's name associated with them.
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.3$. Series
- 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) ... (previous) ... (next): $\S 1.2$: Infinite Series of Constants
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): converge (series)