i have three questions:(adsbygoogle = window.adsbygoogle || []).push({});

1) find the limit of [tex]b_n=\frac{1}{\sqrt n^2}+\frac{1}{\sqrt(n^2-1)}+...+\frac{1}{\sqrt(n^2-(n-1)^2)}[/tex]

2) if a is any number greater than -1, evaluate [tex]\lim_{n\rightarrow\infty} \frac{1^a+2^a+...+n^a}{n^{a+1}}[/tex]

3) prove that [tex]\int_{0}^{x}[\int_{0}^{u}f(t)dt]du=\int_{0}^{x}f(u)(x-u)du[/tex]

for the first i got: half pi, and for the second question i got 1/(a+1) is this correct?

for the third question, here's what i did:

[tex]\int_{0}^{x}u'[\int_{0}^{u}f(t)dt]du=[\int_{0}^{u}f(t)dtu]_{0}^{x}-\int_{0}^{x}uf(u)du[/tex] now my question is can i use here a change of dummy variable here for the first integral, from f(t)dt to f(u)du and to get the equality?

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# Integrals and limits.

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