Summations

Examples > s.a. series [including Taylor series].

* Simple powers:

$\sum_{k=1}^n k = { \textstyle{1\over2}}\,n(n+1)\;,\qquad \sum_{k=1}^n k^2 = { \textstyle{1\over6}}\,n(n+1)(2n+1)\;,\qquad \sum_{k=1}^n k^3 = { \textstyle{1\over4}}\,n^2(n+1)^2\;.$

* Inverse powers:

$\sum_{k=1}^\infty {1\over k^2} = \zeta(2) = {1\over3}\,\psi_1(1/2) = {\pi^2\over6} \quad{\rm(the\ Basel\ problem)\;,}\\ \sum_{k=1}^\infty {1\over k\,(k+1)} = 1\;,\qquad \sum_{k=1}^\infty {1\over k\,(k+1)\,(k+2)} = {1\over4}\;, \qquad \sum_{k=1}^\infty {1\over k\,(k+1)\ldots(k+p)} = {1\over p\,(p!)}\;.$

* Geometric and related sums:

$\sum_{i=m}^n q^i = {q^m-q^{n+1}\over1-q}\;, \qquad \sum_{i=0}^n (i+1)\,q^i = {1 - (n+2)q^{n+1} + (n+1)q^{n+2}\over1-q}\;.$

* Binomial:

$\matrix{ {\displaystyle\sum_{k=0}^n {n\choose k} = 2^n}\;,\hfil &{\displaystyle\sum_{k=0}^n \,(-1)^k {n\choose k} = 0}\;,\hfil \cr {\displaystyle\sum_{k=0}^n k\,{n\choose k} = n\,2^{n-1}}\;,\hfil &{\displaystyle\sum_{k=0}^n \,(-1)^k k\,{n\choose k} = 0}\;,\hfil \cr {\displaystyle\sum_{k=0}^n {n\choose k}^{\!2} = {2n\choose n}}\;,\hfil &{\displaystyle\sum_{k=0}^m {n+k\choose n} = {n+m+1\choose n+1}\;,}\hfil \cr {\displaystyle\sum_{k=0}^m {k\choose n} = {m+1\choose n+1}}\;,\qquad\hfil &{\displaystyle\sum_{k=0}^m k\,{k\choose m} = (n+1)\,{m+1\choose n+2} + n\,{m+1 \choose n+1}}\;.\hfil }$

@ References: CRC tables; Valean 19 [derivations and difficult cases].
@ Types of sums: Datta & Agrawal Math(17)-a1609 [trigonometric]; Silagadze MI(19)-a1908 [Basel problem].