The problem is, as the title suggests, to find the Power Series Expansion of $\frac{1}{1- \cos x}$ around $x=c$.
What I've tried:
- Direct Computation: Derivatives get very ugly quickly, and don't yield a nice formula that I can recognize as a "series."
- Tried finding the integral of $\frac{1}{1- \cos x}$, finding it's series and then differentiating it to get the new series.
- Tried reverse of the above, differentiating and finding it's series, then integrating (very messy).
- Then I tried some substitution "tricks", like using the series of $\frac{1}{1-x}$ and then plugging in the series expansion for $\cos x$, but that's a double sum that I struggled to produce anything useful from :$\displaystyle \sum_{k=0}^\infty\left(\sum_{n=0}^\infty\frac{(-1)^nx^{2n}}{(2n)!}\right)^k$
I am literally at my witts end with this problem. I have spent perhaps a day or two trying to figure it out, because I feel that I am so close - but just barely missing something. I do not want the solution posted - now it's personal and I have to figure it out, but I would greatly appreciate a hint in the right direction, or to point out a mistake that I may be overlooking.