I have the following infinite series:
How can I check whether this infinite series is convergent or diverging?
Hint. You may find it more pleasurable to apply the root test, which is essentially equivalent to using user17762's asymptote or Coffee_Table's methodology.
Do you want another idea? Good:
Since the rightmost expression is one belonging to a geometric sequence with common quotient less than one its series converges and thus ours does as well.