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