Examples 4.2.14(a):
To apply the root test, we have to checklim sup = lim sup 1 / = 1by our result on the limit of the n-th root sequence. Hence, the root test, applied to the above series, is inconclusive.
However, since this series is the harmonic series, we can prove either directly that it diverges or apply the p-series test to show divergence.
Note that the root test also does not apply to the alternating harmonic series, which turns out to converge conditionally.