Prove using Limit Theorems

Question

using limit to show that the sequence $$\left\{\frac{(-1)^n\cdot n}{2n-1}\right\}$$ is diverges.

pf:$$\frac{(-1) n^{\frac1n}}{(2n-1)^{\frac1n}}$$ with $\lim$ from $n \to\infty$, I have $-1$.

I don't think that is diverges...help plz.

Answer 1

If you know the following claim, then this will be helpful for you.

$\lim_{n\to \infty}a_n=A \iff \lim_{n\to \infty}a_{2n}=\lim_{n\to \infty}a_{2n+1}=A.$

Clearly, $$\lim_{n\to \infty}a_{2n}=\frac12\not=-\frac12=\lim_{n\to \infty}a_{2n+1}$$

Therefore $\lim_{n\to \infty}a_n$ diverges.

Answer 2

rewrite the given term in the form $(-1)^n\frac{1}{2-\frac{1}{n}}$ thus the sequence diverges