Does
series $\sum\limits_{n=2}^{\infty}\left(\sqrt{n+3}-\sqrt{n+2}\right)$ converge ?
I tried by multiplying it with the conjugate , but can I do that as both tends to infinity can we assure that the difference of these two terms are not zero?
I don't know how to type the exact thing , but cananyone please be kind enough to edit it?
Thank you so much !
Best Answer
Hint: First find the finite sum $$\sum\limits_{n=2}^{N}\left(\sqrt{n+3}-\sqrt{n+2}\right),$$ then take the limit as $N\to\infty$ to find your answer. To find this finite sum, try writing out several terms and you should see that most terms cancel out. (It is a telescopic series.)