MHB Use the Ratio Test for Convergence/Divergence

[Fernando Revilla]

I quote a question from Yahoo! Answers

Use the ratio test to solve:? 1) k=1--> inf (1-(1/k))^(3k) 2)Is it convergent or divergent?

I have given a link to the topic there so the OP can see my response.

 

[Fernando Revilla]

We have

$\displaystyle\lim_{k\to +\infty}\frac{u_{k+1}}{u_k}=\lim_{k\to +\infty}\frac{\left(1-\frac{1}{k+1}\right)^{3k+3}}{\left(1-\frac{1}{k}\right)^{3k}}=\lim_{k\to +\infty}\left(1-\frac{1}{k+1}\right)^3\cdot\lim_{k\to +\infty}\frac{\left(1-\frac{1}{k+1}\right)^{3k}}{\left(1-\frac{1}{k}\right)^{3k}}$

$=1\cdot\displaystyle\lim_{k\to +\infty}\left(\frac{k^2}{k^2-1}\right)^{3k}=\ldots=1\cdot 1=1\text{ (inconclusive)}$

But $\displaystyle\lim_{k\to +\infty}u_k=\displaystyle\lim_{k\to +\infty}\left(1-\frac{1}{k}\right)^{3k}=\ldots=e^{-3}\ne 0$ so, (term test) the series is divergent.