Question

The fourth term in Taylor series of $\log x$ centered at $a=1$ is?

• A. $\frac{(x-1{)}^3}{3}$
• B. $\frac{(x-1{)}^2}{2}$
• C. $-\frac{(x-1{)}^4}{4}$
• D. $(x-1)$

Answer 1

The correct option is C $-\frac{(x-1{)}^4}{4}$

The taylor series expansion of $\log x$ is $f\left(x\right)=\ln \left(x\right)$

$=(x-1)-\frac{1}{2}{(x-1)}^2+\frac{1}{3}{(x-1)}^3-\frac{1}{4}{(x-1)}^4+\cdots$ $f\left(x\right)$

$=\sum _{n=1}^{\infty }\left[\frac{{(-1)}^{n+1}}{n}{(x-1)}^n\right]$