# convergence of complex term series

A series

 $\displaystyle\sum_{\nu=1}^{\infty}c_{\nu}\;=\;c_{1}\!+\!c_{2}\!+\!c_{3}\!+\ldots$ (1)

with complex terms

 $c_{\nu}\;=\;a_{\nu}\!+\!ib_{\nu}\qquad(a_{\nu},\,b_{\nu}\in\mathbb{R}\;\;% \forall\,\nu)$

is convergent iff the sequence of its partial sums converges to a complex number.

Theorem 1.  The series (1) converges iff the series

 $\displaystyle\sum_{\nu=1}^{\infty}a_{\nu}\quad\mbox{and}\quad\sum_{\nu=1}^{% \infty}b_{\nu}$ (2)

formed by real parts and the imaginary parts of its terms both are convergent.

Proof.  Let  $\varepsilon>0$.  Denote

 $\sum_{\nu=1}^{n}a_{\nu}\,:=\,s_{n},\quad\sum_{\nu=1}^{n}b_{\nu}\,:=\,t_{n},% \quad\sum_{\nu=1}^{n}c_{\nu}\,:=\,u_{n}.$

If the series (2) are convergent with sums $S$ and $T$, then there is a number $N$ such that

 $|s_{n}-S|<\frac{\varepsilon}{2},\quad|t_{n}-T|<\frac{\varepsilon}{2}\quad\mbox% {when}\quad n\geqq N.$

Accordingly,

 $|u_{n}-(S\!+\!iT)|=\sqrt{(s_{n}-S)^{2}+(t_{n}-T)^{2}}\leqq|s_{n}-S|+|t_{n}-T|<% \varepsilon\quad\mbox{when}\quad n\geqq N,$

i.e. the series (1) converges to $S\!+\!iT$.  If, conversely, (1) converges to a complex number

 $u\,=\,s\!+\!it\quad(s,\,t\in\mathbb{R}),$

then

 $|s_{n}-s|\,\leqq\,|(s_{n}-s)+i(t_{n}-t)|\,=\,|u_{n}-u|,\quad|t_{n}-t|\,\leqq\,% |(s_{n}-s)+i(t_{n}-t)|\,=\,|u_{n}-u|,$

and consequently,  $\displaystyle\lim_{n\to\infty}s_{n}\,=\,s$  and  $\displaystyle\lim_{n\to\infty}t_{n}\,=\,t$, i.e. the series (2) are convergent with sums the real numbers $s$ and $t$.

Theorem 2.  The series (1) converges absolutely iff the series (2) both converge absolutely.

Proof.  The absolute convergence of (1) means that the series

 $\sum_{\nu=1}^{\infty}|c_{\nu}|$

converges.  But since  $|c_{\nu}|^{2}\,=\,|a_{\nu}|^{2}+|b_{\nu}|^{2}$,  we have

 $|a_{\nu}|\,\leqq\,|c_{\nu}|;\quad|b_{\nu}|\,\leqq\,|c_{\nu}|\,\leqq\,|a_{\nu}|% +|c_{\nu}|.$

From these inequalities we can infer the assertion of the theorem 2.

## References

• 1 R. Nevanlinna & V. Paatero: Funktioteoria.  Kustannusosakeyhtiö Otava. Helsinki (1963).
Title convergence of complex term series ConvergenceOfComplexTermSeries 2014-10-31 19:04:59 2014-10-31 19:04:59 pahio (2872) pahio (2872) 8 pahio (2872) Theorem msc 30A99 msc 40A05 OrderOfFactorsInInfiniteProduct