How to show that $$\sum_{n=1}^\infty \bigg(\frac{\sqrt{n+2}-\sqrt{n}}{\sqrt{n+1}-\sqrt{n}}\bigg)$$ is divergent? I tried multiplying by the conjugate. I got $$({\sqrt{n+2}-\sqrt{n}})({\sqrt{n+1}+\sqrt{n}})$$ How to prove that it is divergent? Is there any way to prove it by comparison test?
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3The terms of the series have to go to zero. Does this happen? – Umesh Shankar Jan 05 '23 at 13:26
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It doesn't. Thank you. I completely forgot to always check the necessary condition for the convergence. – Adrianna Jan 05 '23 at 13:42
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Since you have $$ \lim_{n\to\infty}\frac{\sqrt{n+2}-\sqrt{n}}{\sqrt{n+1}-\sqrt n}=2 $$ and $2\neq0$, your series diverges.
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yes... the necessary condition for the convergence... I forgot to firstly check it... Thank you! – Adrianna Jan 05 '23 at 13:43
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