Various formulae
Sums of powers
First three
\[1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2} \\
1^2 + 2^2 + 3^2 + \cdots + n^2 = \frac{n(n+1)}{2} \cdot \frac{(2n+1)}{3} \\
1^3 + 2^3 + 3^3 + \cdots + n^3 = \left[\frac{n(n+1)}{2}\right]^2\]
General
\[\sum _{k=1}^{n}k^{p}={\frac {n^{p+1}}{p+1}}+{\frac {1}{2}}n^{p}+\sum _{k=2}^{p}{\frac {B_{k}}{k!}}\left(\frac{p!}{p-k+1}\right)n^{p-k+1}\]
Geometric series
\[\sum_{k=0}^{n} x^k = \frac{x^{n+1}-1}{x-1}\]
