A series

(1)

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

and

(2)

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 $$|s_n-S| < \frac{\varepsilon}{2}, \quad |t_n-T| < \frac{\varepsilon}{2} \quad \mbox{when} \quad n \geqq N.$$ Consequently, $$|u_n-(S\!+\!iT)| = \sqrt{(s_n-S)^2+(t_n-T)^2} \leqq |s_n-S|+|t_n-T| < \varepsilon \quad \mbox{when} \quad n \geqq N,$$ 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.

Bibliography

1

R. NEVANLINNA & V. PAATERO: Funktioteoria. Kustannusosakeyhtiö Otava. Helsinki (1963).