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Thread: methods on determining convergence or divergence

  1. #1
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    methods on determining convergence or divergence

    Can you use either taking the limit or solve the integral to find out if $\displaystyle \sum_{n=1}^\infty\frac{n}{\sqrt{n^2+1}}$ diverges or converges?
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Possible actuary View Post
    Can you use either taking the limit or solve the integral to find out if $\displaystyle \sum_{n=1}^\infty\frac{n}{\sqrt{n^2+1}}$ diverges or converges?
    the limit test for divergence is the easiest method.

    $\displaystyle \lim_{n \to \infty} \frac {n}{ \sqrt {n^2 + 1}} \neq 0$ thus the series diverges
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  3. #3
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    Quote Originally Posted by Possible actuary View Post
    Can you use either taking the limit or solve the integral to find out if $\displaystyle \sum_{n=1}^\infty\frac{n}{\sqrt{n^2+1}}$ diverges or converges?
    The problem becomes harder if we consider $\displaystyle \sum_{n=1}^\infty\frac{n}{\sqrt{n^3+1}}$ or $\displaystyle \sum_{n=1}^\infty\frac{n}{\sqrt{n^4+1}}$
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