TITLE: Power series of $\ln x$ near $2$
QUESTION [5 upvotes]: I wanted to find the Taylor series of $f(x)=\ln x$ around $2$, is below correct?
Observe that:
(1).$f(2)=\ln 2$ and $f(x)=\ln 2$
(2).$f'(2)=\frac 12$ and $f'(x)=\frac 1x$
(3). $f''(2) =-1\times 2^{-2}$ and $f''(x)=-x^{-2}$
(4). $f'''(2)=-1\times-2\times2^{-3}$ and $f'''(x)=2x^{-3}$
So by induction, in general we get :
$f^{(n)}(2)=(-1)^{n-1}2^{-n}(n-1)! $
And the series is  $\ln 2+ \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{2^nn}(x-2)^n$
And is there any other ways to get the series?

REPLY [2 votes]: There are other ways to get the series as well, for example consider
\begin{align*}
\ln x & = \int_1^x \frac{1}{t} \, dt\\
& = \int_1^x \frac{1}{2+t-2} \, dt\\
& =\frac{1}{2} \int_1^x \frac{1}{1+\frac{t-2}{2}} \, dt\\
\end{align*}
Now use geometric series to expand the integrand and then integrate term wise to get the series.