how to evaluate the product $\prod _{n=2}^\infty (1+ \frac{1}{n^2}+\frac{1}{n^4}+\frac{1}{n^6}+\cdots )$? Evaluating the infinite  product of $\prod _{n=2}^\infty (1+ \frac{1}{n^2}+\frac{1}{n^4}+\frac{1}{n^6}+\cdots )$. Please Help.

Step $1$: Notice that $$\prod_{n=2}^{\infty} \bigg(\sum_{k=0}^{\infty} \frac{1}{n^{2k}}\bigg) = \prod_{n=2}^{\infty} \frac{1}{1-\frac{1}{n^2}} = \prod_{n=2}^{\infty}\frac{n^2}{n^2-1}$$
Step $2$: Use induction on $N$ to show that $$\prod_{n=2}^N \frac{n^2}{n^2-1} = \frac{2N}{N+1}$$
Step 3: Conclude that $$\prod_{n=2}^{\infty} \frac{n^2}{n^2-1}  = \lim_{N \to \infty} \frac{2N}{N+1} = 2$$
