I am trying to evaluate the following limit:
$$lim_{n \rightarrow \infty} \sum_{k=1}^{n} {\frac{1}{(n+k)^\frac{1}{ n}}}$$
At first I thought that, since this is an infinite limit of a finite sum (partial sums), it would just be the infinite series of the inner term - and so I would need to do the convergence tests - but I'm not sure.
I also thought of exchanging the limit with the sum (evaluate the limit of the term inside the sum first and then sum it) - but I know it has to do with measure theory (Dominated Convergence Theorem), but I don't know so much about that theory.
If you have any ideas on what's the right way to approach this, please help.