I need help finding sup and inf of the following sequence, and determine whether the sequence has a maximum or a minimum.
$$\bigg\{ \left(1+ \frac{(-1)^n}{2n} \right)^n \bigg\}^\infty_{n=1}$$
The subsequence for even n converges, I think:
$$\bigg\{ \left(1+ \frac{1}{2k} \right)^k \bigg\}^\infty_{k=2} \quad k = 2n$$ $$\bigg\{ \sqrt{\left(1+ \frac{1}{2k} \right)^{2k}} \bigg\}^\infty_{k=2p}$$ to $\sqrt{e}$. Similliar for the odd: $$ \bigg\{ \left[\left(1+ (\frac{1}{-2p} )\right)^{-2p}\right]^{-1/2} \bigg\}^\infty_{p=2n-1} \to e^{-1/2}= \dfrac{1}{\sqrt{e}} \quad \text{when p}\to\infty$$
However, I don't really know what this says. Can I apply Bolzano-Weierstrass theorem? Is the original sequence equal to the two subsequences? As you can tell I need some guidance. Thank you in advance!
Hint : The even subsequence is strictly increasing and convergent to $\sqrt{e}$. The odd subsequence is strictly increasing and convergent to $1/\sqrt{e}$.
Moreover the term for $n=1$ is equal to $1/2$, and the term for $n=2$ is equal to $25/16$.
With all these informations, you should be able to answer the question.