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?