Series of Nonnegative Real Numbers, Comparison, and the Geometric Series

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 convergence of a series and its sum be as defined there. Then the following hold.

1. (Criterion for nonnegative terms)(i) Suppose $0\le a_{k}$ for every $k\in\mathbb{N}$. Then $s_{n}\le s_{n+1}$ for every $n\in\mathbb{N}$, and $\sum_{k=1}^{\infty}a_{k}$ converges if and only if the set $\{s_{n}:n\in\mathbb{N}\}$ is bounded above; in that case

$$\sum_{k=1}^{\infty}a_{k}=\sup\{s_{n}:n\in\mathbb{N}\}.$$

2. (Domination by the sum)(ii) Suppose $0\le a_{k}$ for every $k\in\mathbb{N}$ and $\sum_{k=1}^{\infty}a_{k}$ converges. Then

$$0\le s_{n}\le\sum_{k=1}^{\infty}a_{k}\qquad\text{for every }n\in\mathbb{N}.$$

3. (Comparison test)(iii) Suppose $0\le a_{k}\le b_{k}$ for every $k\in\mathbb{N}$ and that $\sum_{k=1}^{\infty}b_{k}$ converges. Then $\sum_{k=1}^{\infty}a_{k}$ converges and

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

4. (Geometric series)(iv) Let $r\in\mathbb{R}$ satisfy $0\le r<1$, and let $r^{k}$ denote the natural power. Then $1-r$ is positive, the sequence $(r^{k})_{k\in\mathbb{N}}$ converges to $0$, and

$$\sum_{k=1}^{n}r^{k}=\frac{r-r^{n+1}}{1-r}\qquad\text{for every }n\in\mathbb{N}.$$

The series $\sum_{k=1}^{\infty}r^{k}$ converges, and for every $n\in\mathbb{N}$

$$\sum_{k=1}^{\infty}r^{k}=\frac{r}{1-r}, \qquad \sum_{k=1}^{\infty}r^{k}-\sum_{k=1}^{n}r^{k}=\frac{r^{n+1}}{1-r}.$$

In particular, taking $r=\tfrac{1}{2}$, the series $\sum_{k=1}^{\infty}\bigl(\tfrac{1}{2}\bigr)^{k}$ converges with sum $1$, and $\sum_{k=1}^{\infty}\bigl(\tfrac{1}{2}\bigr)^{k}-\sum_{k=1}^{n}\bigl(\tfrac{1}{2}\bigr)^{k}=\bigl(\tfrac{1}{2}\bigr)^{n}$ for every $n\in\mathbb{N}$.

5. (Tail bound for a dominated series of nonnegative terms)(v) Let $(\mu_{k})_{k\in\mathbb{N}}$ be a sequence of nonnegative real numbers such that $\sum_{k=1}^{\infty}\mu_{k}$ converges, let $M\in\mathbb{R}$ be nonnegative, and let $(w_{k})_{k\in\mathbb{N}}$ be a sequence of real numbers with $0\le w_{k}\le M$ for every $k\in\mathbb{N}$. Then the series $\sum_{k=1}^{\infty}\mu_{k}w_{k}$ converges and, for every $n\in\mathbb{N}$,

$$0\le\sum_{k=1}^{\infty}\mu_{k}w_{k}-\sum_{k=1}^{n}\mu_{k}w_{k}\le M\Bigl(\sum_{k=1}^{\infty}\mu_{k}-\sum_{k=1}^{n}\mu_{k}\Bigr).$$