Finding the Taylor series for $\ln(\sin(x))$ I'm trying to calculate the Taylor series for $\ln(\sin(x))$, though I'm not sure how to correctly expand the series.
when $a = \frac{\pi}{4}$
Here's what I have so far:
$$\ln(\sin(x))= \ln\left(x-\frac{x^3}{3!}+ \cdots+ (-1)^{n-1}\frac{x^{2n-2}}{(2n-2)!}\right)$$
What are some tips for expanding logarithmic series, for I have many exercises very similar to this and need an approach, though I'm stuck on converting them into series, your help is much appreciated!
As far as I'm aware, I've noticed two approaches thus far.

*

*$\frac{dy}{dx}=\frac{1}{\sin(x)}\cdot \cos(x) = \cot(x) \implies \int\cot(x)$
Given the taylor series is:
If we let $(x-a) = h$
$f(a+h) = f(a) + f'(a)h+\frac{f''(a)}{2!}h^2 ...$
Then the first few series should be:
$\ln(\sin(x)) = \ln(\sin(\frac{\pi}{4}))+\cot(\frac{\pi}{4})(x-\frac{\pi}{4})...$?
 A: There can be two approaches here, in both, you shall have to make direct use of Taylor's formula:
\begin{align*}f\left( x \right) & = \sum\limits_{n = 0}^\infty {\frac{{{f^{\left( n \right)}}\left( a \right)}}{{n!}}{{\left( {x - a} \right)}^n}} \\ & = f\left( a \right) + f'\left( a \right)\left( {x - a} \right) + \frac{{f''\left( a \right)}}{{2!}}{\left( {x - a} \right)^2} + \frac{{f'''\left( a \right)}}{{3!}}{\left( {x - a} \right)^3} + \cdots \end{align*}
Method-1:
Take $ f(x) = ln(y) $ with $ y= \sin (x)$ and $ a = \sin(\pi / 4)$ and expand directly
Method-2:
Make use of the fact that:
$$\frac{d}{dx} \ln (\sin x ) =\frac{1}{\sin x} \cdot \cos x = \cot (x)$$
So, you can directly evaluate with $f(x) = \cot x $ and $ a = \pi / 4 $ and integrate from $0$ to $x$

I trust that you will be able to perform the calculation now.
A: The problem there is you are still talking the logarithm of an infinite series, so its not actually a Taylor series as such, instead you would need to derive the Taylor series from the start using the general formula for Taylor series. Another problem you will encounter is that $\sin(x)=0$ has infinite repeating solutions due to its periodicity and so your function will also be undefined at all of these points.
