root test
The root test asks: what is the typical per-term scaling factor, extracted by taking the n-th root of each term's size? If that scaling factor is eventually below one, the terms decay geometrically and the series converges; if it exceeds one, the terms do not even tend to zero and the series diverges. It is the natural test when a term is an n-th power.
Formally, set L = limsup |a_n|^{1/n} (the limit superior, which always exists in the extended reals). Then: if L < 1 the series converges absolutely; if L > 1 the series diverges; if L = 1 the test is inconclusive. Using the limit superior rather than the plain limit makes the root test apply even when |a_n|^{1/n} oscillates, which is one reason it is theoretically the sharpest of the elementary ratio-type tests.
A precise comparison with the ratio test: whenever the ratio test gives a verdict, the root test gives the same one, but there are series the root test decides and the ratio test cannot. The price is that n-th roots are sometimes harder to compute than ratios. Like the ratio test, it is powerless exactly at L = 1, where p-series again form the boundary examples.
For sum (n/(2n+1))^n, |a_n|^{1/n} = n/(2n+1) -> 1/2 < 1, so the series converges. For sum (1 + 1/n)^{-n^2} the root test cleanly handles the n-th power.
When terms are n-th powers, taking the n-th root is the natural move.