| Series Name | Series Description | Comments |
|---|---|---|
| Geometric series (p. 557) | \(\sum\limits_{k=1}^{\infty }ar^{k-1}=a+ar+ar^{2} +\cdots ,~a\neq 0\) | Converges to \(\dfrac{a}{1-r}\) if \(\vert r\vert <1\); diverges if \(\vert r\vert \geq 1.\) |
| Harmonic series (p. 561) | \(\sum\limits_{k=1}^{\infty }\dfrac{1}{k}=1+\dfrac{1}{2}+\dfrac{1}{3}+\cdots \) | Diverges. |
| p-series (p. 570) | \(\sum\limits_{k=1}^{\infty }\dfrac{1}{k^{p}}=1+\dfrac{1}{2^{p}}+\dfrac{1}{3^{p}}+\cdots \) | Converges if \(p>1\); diverges if \(0<p\leq 1\). |
| \(k\)-to-the-\(k\) series (p. 576) | \(\sum\limits_{k=1}^{\infty }\dfrac{1}{k^{k}}=1+\dfrac{1}{2^{2}}+\dfrac{1}{3^{3}}+\dfrac{1}{4^{4}}+\cdots \) | Converges. |
| Factorial series (p. 591) | \(\sum\limits_{k\,=\,0}^{\infty }\dfrac{1}{k!}=1+1+\dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{24}+\cdots \) | Converges. |
| Alternating harmonic series (p. 583) | \(\sum\limits_{k=1}^{\infty }\dfrac{(-1)^{k+1}}{k}=1-\dfrac{1}{2}+\dfrac{1}{3}- \dfrac{1}{4}+\cdots \) | Converges. |