Determine the domain of convergence for the power series
{sum infinity with n=0}$\displaystyle (e^n/(n^5+1))(x/2)^n$
Where are you stuck here? I might have made an error somewhere along the way, but it looks to me that if you apply the basic formula or use the ratio test - carefully and putting in all the steps - you get |x| > 2e.
So the only question left is whether there's convergence at x = 2e and x = -2e .....
The algebra in getting |x| > 2e is not difficult, just quite awkward. So you do it ..... carefully and putting in all the steps. Perhaps post your working so it can be seen where you're stuck.
If it's testing convergence at the endpoints, please say so.