Example:
The root test does not apply to
,
but the series converges.
To apply the root test, we have to compute the limit superior of the
expression
By our result on the limit of the n-th root sequence, the
lim sup of this expression is equal to 1. Hence, the root test
is inconclusive.
However, the above series is a p-series with p = 2, and
therefore converges by the p-series test.
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