Question

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

Answer 1

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.