Question

Does the sum converge or diverge? \[\sum_{n=1}^{\infty}(-1)^n3^{\frac{2}{n}}\]
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The answer says \[\lim_{n \rightarrow \infty}3^{\frac{2}{n}} \rightarrow 3^0 \rightarrow 1\]
and that (-1)^n doesn't affect the convergence.

Why doesn't (-1)^n affect the convergence? Maybe I'm wondering because I now \[1^\infty\] is indeterminate... so what's \[(-1)^\infty\]?

Answer 1

I know*

Answer 2

if \[\sum_{n=1}^{\infty}|a_n|\] converges then \[a_n\to0\]

the contrapositive

if \[\lim_{n\to\infty}a_n\neq0\text { or DNE}\]
then \[\sum_{n=1}^{\infty}|a_n|\] diverges