Elementary Properties of Series of Real Numbers: Linearity, Null Terms, the Cauchy Criterion, Index Shifts, Tails, Order and Telescoping

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, let $(a_{k})$, $(b_{k})$ and $(c_{k})$ be sequences in $\mathbb{R}$, let $s_{n}$ and $t_{n}$ be the $n$-th partial sums of $(a_{k})$ and $(b_{k})$, let convergence of a series and its sum be as in Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence §converges, and let convergent and null sequences and limits be as in Convergent Sequences of Real Numbers §converges and The Limit of a Convergent Sequence §limit, and let finite sums over intervals be as in Sums and Products over a Finite Set and over an Interval §intervals and Sums and Products over a Finite Set and over an Interval §sums. For $q\in\mathbb{N}$, $\sum_{k=1}^{\infty}a_{q+k}$ is the series of the sequence $(a_{q+k})_{k\in\mathbb{N}}$. Let $\lambda\in\mathbb{R}$ and $p\in\mathbb{N}$.

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

¶ If $\sum_{k=1}^{\infty}a_{k}$ converges, then $(a_{k})$ is a null sequence.

¶ $\sum_{k=1}^{\infty}a_{k}$ converges if and only if for every real $\varepsilon>0$ there is $N\in\mathbb{N}$ with $\big|\sum_{k=m+1}^{n}a_{k}\big|<\varepsilon$ for all $m,n\in\mathbb{N}$ with $N\le m<n$; and each of these two conditions is equivalent to $(s_{n})$ being a Cauchy sequence.

¶ $\sum_{k=1}^{\infty}a_{k}$ converges if and only if $\sum_{k=1}^{\infty}a_{p+k}$ converges, and then $\sum_{k=1}^{\infty}a_{k}=\sum_{k=1}^{p}a_{k}+\sum_{k=1}^{\infty}a_{p+k}$.

¶ If $\sum_{k=1}^{\infty}a_{k}$ converges, then $\sum_{k=1}^{\infty}a_{q+k}$ converges for every $q\in\mathbb{N}$, and $\big(\sum_{k=1}^{\infty}a_{q+k}\big)_{q\in\mathbb{N}}$ is a null sequence.

¶ If $\sum_{k=1}^{\infty}a_{k}$ and $\sum_{k=1}^{\infty}b_{k}$ converge and $a_{k}\le b_{k}$ for every $k\in\mathbb{N}$, then $\sum_{k=1}^{\infty}a_{k}\le\sum_{k=1}^{\infty}b_{k}$.

¶ If $a_{k}=c_{k+1}-c_{k}$ for every $k\in\mathbb{N}$, then $s_{n}=c_{n+1}-c_{1}$ for every $n\in\mathbb{N}$; moreover $\sum_{k=1}^{\infty}a_{k}$ converges if and only if $(c_{k})$ is convergent, and then $\sum_{k=1}^{\infty}a_{k}=\lim_{k\to\infty}c_{k}-c_{1}$.