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

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- Jan 21st 2007, 10:00 AMvio_viosequence 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 - Jan 21st 2007, 10:38 AMThePerfectHacker
I did not completely work out the details, meaning evaluate the limits.

But note that,

$\displaystyle \sqrt{\frac{n}{n+1}}=\frac{1}{\sqrt{n^2+n}}+...+\f rac{1}{\sqrt{n^2+n}} \leq \frac{1}{\sqrt{n^2+1}}+...+\frac{1}{\sqrt{n^2+n}}$

And,

$\displaystyle \frac{1}{\sqrt{n^2+1}}+...+\frac{1}{\sqrt{n^2+n}}\ leq \frac{1}{\sqrt{n^2}}+...+\frac{1}{\sqrt{n^2}}=\fra c{1}{n}+...+\frac{1}{n}=1$

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.