Definition:Convergent Series/Number Field

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Definition

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

Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be a series in $S$.

Let $\sequence {s_N}$ be the sequence of partial sums of $\displaystyle \sum_{n \mathop = 1}^\infty a_n$.

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 $\displaystyle \sum_{n \mathop = 1}^\infty a_n = s$.

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 $\displaystyle \sum_{n \mathop = 1}^\infty a_n$, where:

$a_n = \dfrac {\paren {-1}^n + i \cos n \theta} {n^2}$

is convergent.


Example: $\dfrac 1 {n^2 - i n}$

The series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$, where:

$a_n = \dfrac 1 {n^2 - i n}$

is convergent.


Example: $\dfrac {e^{i n} } {n^2}$

The series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$, where:

$a_n = \dfrac {e^{i n} } {n^2}$

is convergent.


Example: $\paren {\dfrac {2 + 3 i} {4 + i} }^n$

The series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$, where:

$a_n = \paren {\dfrac {2 + 3 i} {4 + i} }^n$

is convergent.


Also see

  • Results about convergence can be found here.
  • Results about 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