boloman0z

2022-06-25

Taylor Series for $\mathrm{log}\left(x\right)$
Does anyone know a closed form expression for the Taylor series of the function $f\left(x\right)=\mathrm{log}\left(x\right)$ where $\mathrm{log}\left(x\right)$ denotes the natural logarithm function?

Xzavier Shelton

Expert

$-\mathrm{log}\left(1-x\right)=x+\frac{{x}^{2}}{2}+\frac{{x}^{3}}{3}+\dots \phantom{\rule{2em}{0ex}}\left(|x|<1\right)$
There is no expansion around $x=1$ because the log is singular at $0$.

Craig Mendoza

Expert

For $x\in \mathbb{R}$ satisfying $0
$f\left(x\right)=\mathrm{ln}\left(x\right)=\left(x-1\right)-\frac{1}{2}{\left(x-1\right)}^{2}+\frac{1}{3}{\left(x-1\right)}^{3}-\frac{1}{4}{\left(x-1\right)}^{4}+\cdots$
$f\left(x\right)=\sum _{n=1}^{\mathrm{\infty }}\left[\frac{{\left(-1\right)}^{n+1}}{n}{\left(x-1\right)}^{n}\right]$

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