Compute $\prod_{n=1}^\infty\bigg(1+\frac{(-1)^n}{n+1}\bigg)$

Question

Hey this is my first time using this website so please fix my formatting if it is bad.

Can someone please help me compute this$$ \prod_{n=1}^\infty\bigg(1+\frac{(-1)^n}{n+1}\bigg) $$

Answer 1

Write out some terms:

$$\frac{1}{2}\cdot \frac{4}{3}\cdot \frac{3}{4}\cdot \frac{6}{5}\cdot \frac{5}{6}\cdots$$

What does that look like?

Answer 2

HINT: an infinity product as $\prod_{k=1}^\infty a_k$ is defined as the limit of the sequence $(b_n)_{n\in\Bbb N}$ defined by the partial products $b_n:=\prod_{k=1}^n a_k$. Now write some $b_n$ and see what the sequence seems to be.

Note that $b_{2n+1}=\frac12$ and $b_{2n}=\frac12\cdot\frac{2n+2}{2n+1}$. If you need a formal proof you can use induction to prove it.