Interactive Real Analysis
- part of Examples 4.2.14(a):
The root test does not apply to
,
but the series diverges.
Back
To apply the root test, we have to check
,
but the series diverges.
Backlim supby our result on the limit of the n-th root sequence. Hence, the root test, applied to the above series, is inconclusive.= lim sup 1 /
= 1
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.