Taylor Series Expansions

\(\huge f(x) = \frac{1}{1+x}\)

\(\huge \frac{1}{1+x}=\sum_0^\infty (-x)^n=1-x+x^2-x^3+x^4...\)
(-1, 1)

…

\(\huge f(x) = e^{x}\)

\(\huge e^{x}=\sum_0^\infty \frac{x^{n}}{n!}=1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+\frac{x^5}{120}...\)
\(x\in \mathbb{R}\)

…

\(\huge ln(1+x)\)
\(\huge \sum_1^\infty ln(1+x)=\frac{(-1)^{n+1}}{n}*x^{n}=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\frac{x^5}{5}-\frac{x^6}{6}...\) (-1, 1]