# Series of Constants

#### Arithmetic Series

$a+(a+d)+(a+2d)+\cdots+\{a+(n-1)d\}=\left(\frac{1}{2}\right)n\{2a+(n-1)d\}=\left(\frac{1}{2}\right)n(a+l)$
where $l=a+(n-1)d$ is the last term.

Some special cases are
$1+2+3+\cdots+n=\left(\frac{1}{2}\right)n(n+1)$
$1+3+5+\cdots+(2n-1)=n^2$

#### Geometric Series

$a+ar+ar^2+ar^3+\cdots+ar^{n}=\frac{a(1-r^{n+1})}{1-r}=a\frac{1-r^{n+1}}{1-r}$
where $r\neq1$.

If $-1< r< 1$, then
$a+ar+ar^2+ar^3+\cdots+ar^n=\frac{a}{1 - r}$

#### Arithmetic - Geometric Series

$a+(a+d)r+(a+2d)r^2+\cdots+\{a+(n-1)d\}r^{n-1}=\frac{a(1-r^n)}{1-r}+\frac{rd\{1-nr^{n-1}+(n-1)r^n\}}{(1-r)^2}$
where $r\neq1$.

If $-1< r< 1$, then
$a+(a+d)r+(a+2d)r^2+\cdots=\frac{a}{1-r}+\frac{rd}{(1-r)^2}$

#### Sums of Powers of Positive Integers

$1^p+2^p+3^p+\cdots+n^p=\frac{n^{p+1}}{p+1}+\frac{1}{2}n^p+\frac{B_1pn^{p-1}}{2!}-\frac{B_2p(p-1)(p-2)n^{p-3}}{4!}+\cdots$
where the series terminates at $n^2$ or $n$ according as $p$ is odd or even, and $B_k$ are the Bernoulli numbers.

Some special cases are

$1+2+3+\cdots+n=\frac{n(n+1)}{2}$

$1^2+2^2+3^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$

$1^3+2^3+3^3+\cdots+n^3=\frac{n^2(n+1)^2}{4}=(1+2+3+\cdots+n)^2$

$1^4+2^4+3^4+\cdots+n^4=\frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}$

If $S_k=1^k+2^k+3^k+\cdots+n^k$ where $k$ and $n$ are positive integers, then
$\binom{k+1}{1}S_1+\binom{k+1}{2}S_2+\cdots+\binom{k+1}{k}S_k=(n+1)^{k+1}-(n+1)$

#### Series Invoving Reciprocals of Powers of Positive Integers

$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\cdots=\ln2$

$1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\cdots=\frac{\pi}{4}$

$1-\frac{1}{4}+\frac{1}{7}-\frac{1}{10}+\frac{1}{13}-\cdots=\frac{\pi\sqrt{3}}{9}+\frac{1}{3}\ln2$

$1-\frac{1}{5}+\frac{1}{9}-\frac{1}{13}+\frac{1}{17}-\cdots=\frac{\pi\sqrt{2}}{8}+\frac{\sqrt{2}\ln(1+\sqrt{2})}{4}$

$\frac{1}{2}-\frac{1}{5}+\frac{1}{8}-\frac{1}{11}+\frac{1}{14}-\cdots=\frac{\pi\sqrt{3}}{9}-\frac{1}{3}\ln2$

$\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\cdots=\frac{\pi^2}{6}$

$\frac{1}{1^4}+\frac{1}{2^4}+\frac{1}{3^4}+\frac{1}{4^4}+\cdots=\frac{\pi^4}{90}$

$\frac{1}{1^6}+\frac{1}{2^6}+\frac{1}{3^6}+\frac{1}{4^6}+\cdots=\frac{\pi^6}{945}$

$\frac{1}{1^2}-\frac{1}{2^2}+\frac{1}{3^2}-\frac{1}{4^2}+\cdots=\frac{\pi^2}{12}$

$\frac{1}{1^4}-\frac{1}{2^4}+\frac{1}{3^4}-\frac{1}{4^4}+\cdots=\frac{7\pi^4}{720}$

$\frac{1}{1^6}-\frac{1}{2^6}+\frac{1}{3^6}-\frac{1}{4^6}+\cdots=\frac{31\pi^6}{30,240}$

$\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+\cdots=\frac{\pi^2}{8}$

$\frac{1}{1^4}+\frac{1}{3^4}+\frac{1}{5^4}+\frac{1}{7^4}+\cdots=\frac{\pi^4}{96}$

$\frac{1}{1^6}+\frac{1}{3^6}+\frac{1}{5^6}+\frac{1}{7^6}+\cdots=\frac{\pi^6}{960}$

$\frac{1}{1^3}-\frac{1}{3^3}+\frac{1}{5^3}-\frac{1}{7^3}+\cdots=\frac{\pi^3}{32}$

$\frac{1}{1^3}+\frac{1}{3^3}-\frac{1}{5^3}-\frac{1}{7^3}+\cdots=\frac{3\pi^3\sqrt{2}}{128}$

$\frac{1}{1\cdot3}+\frac{1}{3\cdot5}+\frac{1}{5\cdot7}+\frac{1}{7\cdot9}+\cdots=\frac{1}{2}$

$\frac{1}{1\cdot3}+\frac{1}{2\cdot4}+\frac{1}{3\cdot5}+\frac{1}{4\cdot6}+\cdots=\frac{3}{4}$

$\frac{1}{1^2\cdot3^2}+\frac{1}{3^2\cdot5^2}+\frac{1}{5^2\cdot7^2}+\frac{1}{7^2\cdot9^2}+\cdots=\frac{\pi^2-8}{16}$

$\frac{1}{1^2\cdot2^2\cdot3^2}+\frac{1}{2^2\cdot3^2\cdot4^2}+\frac{1}{3^2\cdot4^2\cdot5^2}+\cdots=\frac{4\pi^2-39}{16}$

$\frac{1}{a}-\frac{1}{a+d}+\frac{1}{a+2d}-\frac{1}{a+3d}+\cdots=\int\limits_0^1\frac{u^{a-1}\ du}{1+u^d}$

$\frac{1}{1^{2p}}+\frac{1}{2^{2p}}+\frac{1}{3^{2p}}+\frac{1}{4^{2p}}+\cdots=\frac{2^{2p-1}\pi^{2p}B_p}{(2p)!}$

$\frac{1}{1^{2p}}+\frac{1}{3^{2p}}+\frac{1}{5^{2p}}+\frac{1}{7^{2p}}+\cdots=\frac{(2^{2p}-1)\pi^{2p}B_p}{2(2p)!}$

$\frac{1}{1^{2p}}-\frac{1}{2^{2p}}+\frac{1}{3^{2p}}-\frac{1}{4^{2p}}+\cdots=\frac{(2^{2p-1}-1)\pi^{2p}B_p}{(2p)!}$

$\frac{1}{1^{2p+1}}-\frac{1}{3^{2p+1}}+\frac{1}{5^{2p+1}}-\frac{1}{7^{2p+1}}+\cdots=\frac{\pi^{2p+1}E_p}{2^{2p+2}(2p)!}$

#### Miscellaneous Series

$\frac{1}{2}+\cos\alpha+\cos2\alpha+\cdots+\cos n\alpha=\frac{\sin(n+\frac{1}{2})\alpha}{2\sin\frac{\alpha}{2}}$

$\sin\alpha+\sin2\alpha+\sin3\alpha+\cdots+\sin n\alpha=\frac{\sin\left[\frac{1}{2}(n+1)\right]\alpha\sin\frac{1}{2}n\alpha}{\sin\frac{\alpha}{2}}$

$1+r\cos\alpha+r^2\cos2\alpha+r^3\cos3\alpha+\cdots=\frac{1-r\cos\alpha}{1-2r\cos\alpha+r^2}$,   $|r|<1$

$r\sin\alpha+r^2\sin2\alpha+r^3\sin3\alpha+\cdots=\frac{r\sin\alpha}{1-2r\cos\alpha+r^2}$,   $|r|<1$

$1+r\cos\alpha+r^2\cos2\alpha+\cdots+r^n\cos n\alpha=\frac{r^{n+2}\cos n\alpha-r^{n+1}\cos(n+1)\alpha-r\cos\alpha+1}{1-2r\cos\alpha+r^2}$

$r\sin\alpha+r^2\sin2\alpha+\cdots+r^n\sin n\alpha=\frac{r\sin\alpha-r^{n+1}\sin(n+1)\alpha+r^{n+2}\sin n\alpha}{1-2r\cos\alpha+r^2}$

#### The Euler - Maclaurin Summation Formula

$\sum\limits_{k=1}^{n-1}F(k)=\int\limits_0^nF(k)dk-\frac{1}{2}\{F(0)+F(n)\}+\frac{1}{12}\{F'(n)-F'(0)\}-\frac{1}{720}\{F'''(n)-F'''(0)\}+\frac{1}{30,240}\{F^{(v)}(n)-F^{(v)}(0)\}-\frac{1}{1,209,600}\{F^{(vii)}(n)-F^{(vii)}(0)\}$
$+\cdots(-1)^{p-1}\frac{B_p}{(2p)!}\{F^{(2p-1)}(n)-F^{(2p-1)}(0)\}+\cdots$

#### The Poisson Sommation Formula

$\sum\limits_{k=-\infty}^\infty F(k)=\sum\limits_{m=-\infty}^\infty\left\{\ \int\limits_{-\infty}^\infty e^{2\pi imx}F(x)\ dx\right\}$

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