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Math Help - sequence limit

  1. #1
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    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
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  2. #2
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    Quote Originally Posted by vio_vio View Post
    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.
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