Interactive Real Analysis - part of

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Examples 4.2.14(a):

The root test does not apply to , but the series diverges.
To apply the root test, we have to check
lim sup = lim sup 1 / = 1
by 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.

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