Finite series

Powers

$$\sum _{k=1}^{n}k=\frac{n\cdot (n+1)}{2}$$$$\sum _{k=1}^n{k}^2=\frac{n\cdot (n+1)\cdot (2n+1)}{6}$$$$\sum _{k=1}^n{k}^3=\frac{n^2\cdot (n+1{)}^2}{2^2}$$

Binomial coefficients

$$\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})=2^n$$$$\sum _{k=0}^n{(\genfrac{}{}{0ex}{}{n}{k})}^2=(\genfrac{}{}{0ex}{}{2n}{n})$$

Harmonic series

$$1+\frac{1}{2}+...+\frac{1}{n}\approx \ln (n)+0.5772$$

Infinite series

Taylor series

$$f(x)-f(x_k)=(x-x_k)\frac{\partial f}{\partial x}(x_k)+\frac{(x-x_k{)}^2}{2!}\frac{{\partial }^{2}f}{\partial x^2}(x_k)+...$$

TIP

Using Taylor series, we can approximate functions like $e^x$, $\sin (x)$, $\cos (x)$, and $\ln (1+x)$ at $x_k=0$.

Riemann zeta function

$$1+\frac{1}{2^2}+\frac{1}{3^2}+...=\zeta (2)=\frac{{\pi }^2}{6}$$$$1+\frac{1}{2^4}+\frac{1}{3^4}+...=\zeta (4)=\frac{{\pi }^4}{45}$$