How do you use the Nth term test on the infinite series #sum_(n=1)^oorootn(2)# ?

1 Answer
Calculus V.
Aug 11, 2014

Recall that the #n^{th}# term test is a test of divergence only. It states that if the sequence of general terms #{a_n}_{n=1}^\infty# does not converge to 0, then the series #\sum_{n=1}^\infty a_n# is divergent. ATTN: If the limit is 0, nothing can be concluded from this test, and another test needs to be used to decide whether the series converges or not.
In the present case, #a_n=\root[n]2=2^{1/n}#.
Since #\lim_{n\to\infty} 2^{1/n}=2^0=1\ne 0#, we conclude by the #n^{th}# term test that the series #\sum_{n=1}^\infty \root[n]2# is divergent.

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