Summation of logarithmic functions

Question

The sum of series $\frac{(\log3)^1}{1!}+\frac{(\log3)^3}{3!}+\frac{(\log 3)^5}{5!}+\cdots$ is what? Is there a general algorithm to find the summation of logarithms?

Answer 1

The series itself does not have too much to do with logarithms; to see why without getting lost with the $\log$ everywhere, let $\alpha = \log 3$. You want $$ \sum_{n=0}^\infty \frac{(\log 3)^{2n+1}}{(2n+1)!}=\sum_{n=0}^\infty \frac{\alpha^{2n+1}}{(2n+1)!} = \sinh \alpha $$ by the series definition of $\sinh$. That being said, now here we have simplifications because $\alpha=\log 3$. Indeed, recall that, for every $x\in\mathbb{R}$, $$ \sinh x = \frac{e^x-e^{-x}}{2} $$ and therefore here $$ \sinh \log 3 = \frac{e^{\log 3}-e^{-\log 3}}{2} = \frac{3-1/3}{2} = \boxed{\frac{4}{3}}\,. $$