Does the following limit converges for some value of $a,b>0$? How to compute the following limit?
$$\lim_{n\to\infty}\frac{\sum_{j=1}^nj^a\exp(j^b)}{n^a\exp(n^b)}$$
What is the range of possible values of $a,b$ for this to hold?

If $b > 1$ then the limit equals $1$ because the last term of your summation will dominate the sum in the numerator, which is equal to the term in the denominator.
