Question

Determine whether the following sequence converge or diverge. Find the limit if it converges.

Answer 1

\[\sqrt[?]{?}\]\[\sqrt[n]{n^3/3^n}\]

Answer 2

Not sure how to deal with the root. Is it the same as \[\sqrt{n ^{3/n}/3}?\]

Answer 3

That is the n-th root, it is the same as raising to the power (1/n). So you have that
\[\sqrt[n]{\frac{n^3}{3^n}} = \left(\frac{n^3}{3^n}\right)^{\frac{1}{n}} = \frac{n^{\frac{3}{n}}}{3}\]

Answer 4

so the limit would go to 1 and would be inconclusive according to the nth root test

Answer 5

no, the limit goes to 1/3. I see it now

Answer 6

Yes, the limit of the sequence is 1/3.