Series of Real Numbers with Nonnegative Terms, Comparison, Absolute Convergence and the Geometric Series

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, let $(a_{k})$ and $(b_{k})$ be sequences in $\mathbb{R}$, let $s_{n}=\sum_{k=1}^{n}a_{k}$ be the partial sums of $(a_{k})$, and let $r\in\mathbb{R}$. Convergence of a series and its sum are as in Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence §converges, absolute convergence as in Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence §absolute, nondecreasing sequences as in Monotone Sequences §monotone, sequences bounded above as in Bounded Sequences of Real Numbers §bounded, and suprema as in Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order §supremum. For $n\in\mathbb{N}$, $\sum_{k=1}^{\infty}a_{n+k}$ and $\sum_{k=1}^{\infty}b_{n+k}$ are the series of the sequences $(a_{n+k})_{k\in\mathbb{N}}$ and $(b_{n+k})_{k\in\mathbb{N}}$.

¶ If $a_{k}\ge0$ for every $k\in\mathbb{N}$, then $(s_{n})$ is nondecreasing, and $\sum_{k=1}^{\infty}a_{k}$ converges if and only if $(s_{n})$ is bounded above; in that case $\sum_{k=1}^{\infty}a_{k}=\sup\{s_{n}:n\in\mathbb{N}\}$ and $0\le s_{n}\le\sum_{k=1}^{\infty}a_{k}$ for every $n\in\mathbb{N}$.

¶ If $0\le a_{k}\le b_{k}$ for every $k\in\mathbb{N}$ and $\sum_{k=1}^{\infty}b_{k}$ converges, then $\sum_{k=1}^{\infty}a_{k}$ converges and $\sum_{k=1}^{\infty}a_{k}\le\sum_{k=1}^{\infty}b_{k}$; moreover, for every $n\in\mathbb{N}$ the series $\sum_{k=1}^{\infty}a_{n+k}$ and $\sum_{k=1}^{\infty}b_{n+k}$ converge, and $0\le\sum_{k=1}^{\infty}a_{n+k}\le\sum_{k=1}^{\infty}b_{n+k}$.

¶ If $\sum_{k=1}^{\infty}a_{k}$ converges absolutely, then it converges and $\Big|\sum_{k=1}^{\infty}a_{k}\Big|\le\sum_{k=1}^{\infty}|a_{k}|$.

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

¶ If $|r|<1$, then $1-r\neq0$, and the series $\sum_{k=1}^{\infty}r^{k-1}$ and $\sum_{k=1}^{\infty}r^{k}$ converge and converge absolutely, with

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

¶ If $|r|<1$ (so that $1-r\neq0$ and $\sum_{k=1}^{\infty}r^{k}$ converges, by the clause geometric), then for every $n\in\mathbb{N}$,

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