Elementary Properties of Series of Real Numbers

In the setting of The Real Numbers: Standing Notation and Background, let $(a_{k})_{k\in\mathbb{N}}$ and $(b_{k})_{k\in\mathbb{N}}$ be sequences of real numbers, with partial sums $s_{n}$ and $t_{n}$ respectively, and let $\lambda\in\mathbb{R}$. Convergence of a series and its sum are as defined there. Then the following hold.

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

$$\sum_{k=1}^{\infty}(a_{k}+b_{k})=\sum_{k=1}^{\infty}a_{k}+\sum_{k=1}^{\infty}b_{k}, \qquad \sum_{k=1}^{\infty}(\lambda a_{k})=\lambda\sum_{k=1}^{\infty}a_{k}.$$

2. (The terms tend to zero)(ii) If $\sum_{k=1}^{\infty}a_{k}$ converges, then the sequence $(a_{k})_{k\in\mathbb{N}}$ converges to $0$.

3. (Cauchy criterion)(iii) The series $\sum_{k=1}^{\infty}a_{k}$ converges if and only if for every real $\varepsilon>0$ there is $N\in\mathbb{N}$ such that

$$|s_{n}-s_{m}|<\varepsilon\qquad\text{for all }m,n\in\mathbb{N}\text{ with }N\le m\text{ and }N\le n.$$

4. (Comparison of sums)(iv) 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

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

5. (Telescoping series)(v) Let $(c_{k})_{k\in\mathbb{N}}$ be a sequence of real numbers and suppose that $a_{k}=c_{k+1}-c_{k}$ for every $k\in\mathbb{N}$. Then $s_{n}=c_{n+1}-c_{1}$ for every $n\in\mathbb{N}$; consequently $\sum_{k=1}^{\infty}a_{k}$ converges if and only if the sequence $(c_{k})_{k\in\mathbb{N}}$ converges, and in that case

$$\sum_{k=1}^{\infty}a_{k}=\Bigl(\lim_{k\to\infty}c_{k}\Bigr)-c_{1}.$$