Standard Examples of Series: the Harmonic Series, the Reciprocals of the Squares, a Centred Quadratic Sum and the Exponential Series

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, let convergence, divergence and sums of series and absolute convergence be as defined there, and let $k!$, for $k\in\mathbb{N}_{0}$, be the factorial of $k$. Elements of $\mathbb{N}$ and of $\mathbb{N}_{0}$ standing in real expressions are read in $\mathbb{R}$ by The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness §identification, and for $m\in\mathbb{N}$ the difference $m-M-1$ below is formed in $\mathbb{R}$. Let $\alpha$ be a positive real number, let $M\in\mathbb{N}$ and let $x\in\mathbb{R}$. Every denominator below is a positive real number.

¶ The set $\big\{\sum_{m=1}^{n}\frac{1}{m}:n\in\mathbb{N}\big\}$ is not bounded above; hence the series $\sum_{m=1}^{\infty}\frac{1}{m}$ diverges.

¶ $\sum_{m=1}^{M}\frac{1}{m^{2}}\le2-\frac{1}{M}$; the series $\sum_{m=1}^{\infty}\frac{1}{m^{2}}$ converges and its sum is at most $2$.

¶ $\sum_{m=1}^{2M+1}\frac{1}{1+\alpha\,(m-M-1)^{2}}\le1+\frac{4}{\alpha}$.

¶ The series $\sum_{k=1}^{\infty}\frac{x^{k}}{k!}$, $\sum_{k=1}^{\infty}\frac{x^{2k}}{(2k)!}$ and $\sum_{k=1}^{\infty}\frac{x^{2k+1}}{(2k+1)!}$ converge absolutely.