Find-the-sum-n-1-1-n-1-n-n-n-1-

Question Number 11800 by tawa last updated on 01/Apr/17

Find the sum

\underset{n = 1}{\sum^{\infty}} \frac{1}{(n + 1) \sqrt{n} + n \sqrt{n + 1}}

Answered by FilupS last updated on 01/Apr/17

\frac{1}{(n + 1) \sqrt{n} + n \sqrt{n + 1}} = \frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n + 1}}

\Rightarrow \underset{n = 1}{\sum^{\infty}} (\frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n + 1}})

= (\frac{1}{\sqrt{1}} - \frac{1}{\sqrt{2}}) + (\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{3}}) + (\frac{1}{\sqrt{3}} - \frac{1}{\sqrt{4}}) + \dots + (\frac{1}{\sqrt{n - 1}} - \frac{1}{\sqrt{n}}) + (\frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n + 1}})

= \frac{1}{\sqrt{1}} - \frac{1}{\sqrt{n + 1}}

= 1 - \frac{1}{\sqrt{n + 1}}

as n \to \infty, \frac{1}{\sqrt{n + 1}} \to 0

∴= 1

Commented by FilupS last updated on 01/Apr/17

I^{'} m unable to do the working for line 1 .

I used WolframAlpha to expand

Commented by tawa last updated on 01/Apr/17

i appreciate sir . God bless you sir .

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