# sequence limit

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• January 21st 2007, 11:00 AM
vio_vio
sequence limit
Please help me!
I can't find the limit of the following sequence:
[1/sqr(n^2+1) +1/sqr(n^2+2) +1/sqr(n^2+3) +.... 1/sqr(n^2+n)]^n
• January 21st 2007, 11:38 AM
ThePerfectHacker
Quote:

Originally Posted by vio_vio
Please help me!
I can't find the limit of the following sequence:
[1/sqr(n^2+1) +1/sqr(n^2+2) +1/sqr(n^2+3) +.... 1/sqr(n^2+n)]^n

I did not completely work out the details, meaning evaluate the limits.

But note that,
$\sqrt{\frac{n}{n+1}}=\frac{1}{\sqrt{n^2+n}}+...+\f rac{1}{\sqrt{n^2+n}} \leq \frac{1}{\sqrt{n^2+1}}+...+\frac{1}{\sqrt{n^2+n}}$
And,
$\frac{1}{\sqrt{n^2+1}}+...+\frac{1}{\sqrt{n^2+n}}\ leq \frac{1}{\sqrt{n^2}}+...+\frac{1}{\sqrt{n^2}}=\fra c{1}{n}+...+\frac{1}{n}=1$

Now, the limit of the first sequence is 1.
And the limit of the second sequence is also 1.
Thus, this limit is also 1.