Find the sum of arcsin 1/√2 + arcsin (√2 - 1)/√6 + ... + arcsin {√n - √(n - 1)}/√{n(n + 1)}, for n approaching infinity?
I suggest first checking that the series converges.
Calcuating a_n for a few large n suggests that the ratio test would be inconclusive (ratio = 1).
Using Raabe's test for few large n suggests that the series may be divergent. However, while Raabe's test can be used to show that a series is convergent I am not sure that the converse is true.
I suspect that one of the convergence tests in your toolbox may show that this series diverges.