In a calculus book I am reading I have encountered the following problem:
$$\sum_{n=1}^\infty{\frac{n}{(2n+1)!}}$$
The hint is to use Taylor series expansion's for $e^x$. I tried to express the sum as the form $$e^x=\sum_{n=0}^{\infty}{\frac{x^n}{n!}}$$
But I could not find a consistent method, I always end un with different sums of factorials that does not help me solve the problem
The official solution is $$\boxed{\frac{1}{2e}}$$
The excersise is in a chapter that mixes calculus with summation, so the solution will probably include both.
Any help or hint is highly appreaciated! Thanks in advance.
Best Answer
Hint: $\frac{n}{(2n+1)!} = \frac{2n+1-1}{2(2n+1)!} = \frac{1}{2 (2n)!} - \frac{1}{2(2n+1)!}$