Showing a sequence does not converge to a limit.

Question

so ot very familiar with the coding part.

Question: Does this sequence converge to a limit in $R$, if so what limit?

$x_n$=$(-1)^n (\frac{n^2+1}{2n^2-1})$

My claim: Does not converge to a limit.

My justification: Take the limit of this sub sequence $(x_{2n})$ and take the limit of $(x_{2n-1})$, since the limits do not equal each other then the sequence $x_n$ does not have a limit.

Is this correct? Are there any other methods of doing this? Any faster methods? Thank you for help in advance.

Answer 1

Yes, your method is correct.

The odd subsequence converges to $-\frac12$ and the even subsequence converges to $\frac12$. Hence it doesn't converge.