Let $x_n=1+\frac{1}{\sqrt 2}+\frac{1}{\sqrt 3}+.......+\frac{1}{\sqrt n}-2\sqrt{n+1}$
I got to know form comment that $x_n$ is increasing sequence. is there any way to prove limit of sequence.
Any help will be appreciated.
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4Note that $\sqrt{n+2}-\sqrt{n+1}=\frac{1}{\sqrt{n+1}+\sqrt{n+2}}$. Can you continue now? – richrow May 18 '20 at 08:47
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Thanks a lot I got now – Curious student May 18 '20 at 08:53
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@richrow Dear Sir , Is it possible to calculate its limit ? Can you give me some hint – Curious student May 18 '20 at 08:56
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Please don't ask new questions in the comments. Write another question instead, or since no answer has been posted to this one, you can edit it to change the question. – saulspatz May 18 '20 at 09:00
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4Does this answer your question? limit $\lim\limits_{n\to\infty}\left(\sum\limits_{i=1}^{n}\frac{1}{\sqrt{i}} - 2\sqrt{n}\right)$ – the limit is the same as yours because $\sqrt{n+1} - \sqrt n \to 0$. – Martin R May 18 '20 at 09:15
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Also: https://math.stackexchange.com/q/79115, https://math.stackexchange.com/q/2946905 – all found with Approach0 – Martin R May 18 '20 at 09:18