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Show that $\prod(1+a_n)$ converges where $a_n = 1/{\sqrt n} + 1/n$ for n odd, $-1/\sqrt n$ for n even.

I tried to multiplying even number of terms, but it's not cancelled well. All I found is that $(1+a_{2n-1})(1+a_{2n})=\dfrac{2n+\sqrt{2n-1}}{2n+\sqrt{2n}}$. How can I proceed?

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You might note that $ (1+a_{2n-1})(1+a_{2n})=1-\frac1{\sqrt{2n}(\sqrt{2n}+1)(\sqrt{2n}+\sqrt{2n-1})} $ and try to take it from there.