MathCS.org - Real Analysis: Examples 4.2.14(b):

Examples 4.2.14(b):

The root test does not apply to

, but the series converges.

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To apply the root test, we have to compute the limit superior of the expression

(n²)^1/n = n^2/n = (n^1/n)²

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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