Series of Complex Numbers: Absolute Convergence, Comparison, Linearity, Index Shifts, Real and Imaginary Parts, Conjugates, Real Series and the Cauchy Product

In the setting of Complex Analysis: Standing Notation, let convergence, sums and absolute convergence of series in $\mathbb{C}$ be as in Series of Complex Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence (ii) and Series of Complex Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence (iii). Let $(a_{k})$ and $(b_{k})$ be sequences in $\mathbb{C}$, let $(r_{k})$ be a sequence in $\mathbb{R}$, and let $\lambda\in\mathbb{C}$.

(i) Absolute convergence. If $\sum_{k=1}^{\infty}a_{k}$ converges absolutely, then it converges, and

$$\Big|\sum_{k=1}^{\infty}a_{k}\Big|\le\sum_{k=1}^{\infty}|a_{k}|.$$

(ii) Comparison. If $|a_{k}|\le r_{k}$ for every $k\in\mathbb{N}$ and $\sum_{k=1}^{\infty}r_{k}$ converges, then $\sum_{k=1}^{\infty}a_{k}$ converges absolutely, and $\Big|\sum_{k=1}^{\infty}a_{k}\Big|\le\sum_{k=1}^{\infty}r_{k}$.

(iii) Linearity. If $\sum_{k=1}^{\infty}a_{k}$ and $\sum_{k=1}^{\infty}b_{k}$ converge, then $\sum_{k=1}^{\infty}(a_{k}+b_{k})$ and $\sum_{k=1}^{\infty}\lambda a_{k}$ converge, with $\sum_{k=1}^{\infty}(a_{k}+b_{k})=\sum_{k=1}^{\infty}a_{k}+\sum_{k=1}^{\infty}b_{k}$ and $\sum_{k=1}^{\infty}\lambda a_{k}=\lambda\sum_{k=1}^{\infty}a_{k}$.

(iv) Index shifts. Let $p\in\mathbb{N}_{0}$; $\sum_{k=1}^{\infty}a_{p+k}$ is the series of the sequence $(a_{p+k})_{k\in\mathbb{N}}$ (a sequence, as $p+k\in\mathbb{N}$ for $k\in\mathbb{N}$), and for $p=0$ the finite sum $\sum_{k=1}^{0}a_{k}$ is the empty sum $0$ (Sums and Products over a Finite Set and over an Interval (ii)). Then $\sum_{k=1}^{\infty}a_{k}$ converges if and only if $\sum_{k=1}^{\infty}a_{p+k}$ converges, and in that case

$$\sum_{k=1}^{\infty}a_{k}=\sum_{k=1}^{p}a_{k}+\sum_{k=1}^{\infty}a_{p+k}.$$

(v) Real and imaginary parts. $\sum_{k=1}^{\infty}a_{k}$ converges if and only if $\sum_{k=1}^{\infty}\operatorname{Re}a_{k}$ and $\sum_{k=1}^{\infty}\operatorname{Im}a_{k}$ converge, and in that case

$$\operatorname{Re}\sum_{k=1}^{\infty}a_{k}=\sum_{k=1}^{\infty}\operatorname{Re}a_{k},\qquad\operatorname{Im}\sum_{k=1}^{\infty}a_{k}=\sum_{k=1}^{\infty}\operatorname{Im}a_{k}.$$

(vi) Conjugates. If $\sum_{k=1}^{\infty}a_{k}$ converges, then $\sum_{k=1}^{\infty}\overline{a_{k}}$ converges, and

$$\overline{\sum_{k=1}^{\infty}a_{k}}=\sum_{k=1}^{\infty}\overline{a_{k}}.$$

(vii) Real series. $\sum_{k=1}^{\infty}\iota(r_{k})$ converges if and only if $\sum_{k=1}^{\infty}r_{k}$ converges, and then $\sum_{k=1}^{\infty}\iota(r_{k})=\iota\Big(\sum_{k=1}^{\infty}r_{k}\Big)$.

(viii) The Cauchy product. Suppose that $\sum_{k=1}^{\infty}a_{k}$ and $\sum_{k=1}^{\infty}b_{k}$ converge absolutely, and let $(c_{n})$ be the sequence in $\mathbb{C}$ given by

$$c_{n}=\sum_{k=1}^{n}a_{k}\,b_{n+1-k}\qquad(n\in\mathbb{N}),$$

where, for $1\le k\le n$, $n+1-k$ is the difference of $n+1$ and $k$, an element of $\mathbb{N}$ as $k<n+1$ (The Difference of Two Natural Numbers with Zero (i)); $(c_{n})$ is a sequence in $\mathbb{C}$ by Maps from Functional Predicative Relations, and Families of Sections (ii), the condition $y=\sum_{k=1}^{n}a_{k}b_{n+1-k}$ on $n$ and $y$ being predicative by Predicative Formulas for the Vocabulary of Finite Combinatorics, at Any Variable Positions (vi). Then $\sum_{n=1}^{\infty}c_{n}$ converges absolutely, and its sum is

$$\Big(\sum_{k=1}^{\infty}a_{k}\Big)\Big(\sum_{k=1}^{\infty}b_{k}\Big).$$