22: The Root test leads to

Since this constant is smaller than 1, the investigated series and therefore also the given series converge.

The limit from the Ermakoff test can be done relatively easily using the calculations done before for the Root test as applied to the given series.

There are other alternatives. For instance, we know that logarithms grow slower than powers, in particular ln(x) < x/2 for large x. Thus

Or one can change everything into an exponential

and then show that the expression in the exponent goes to negative infinity, there are quite a few ways to get this result.

Anyway, r is less than 1, which confirms convergence of the given series.

Answer