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Math Help - Prove if series converges or diverges

  1. #1
    Senior Member Pinkk's Avatar
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    Prove if series converges or diverges

    \sum\,\frac{n-1}{n^{2}}

    I'm having a brain fart, I don't know if this converges or not.
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by Pinkk View Post
    \sum\,\frac{n-1}{n^{2}}

    I'm having a brain fart, I don't know if this converges or not.
    \sum_{n=1}^{\infty}\frac{n-1}{n^2}=\sum_{n=1}^{\infty}\frac{1}{n}-\sum_{n=1}^{\infty}\frac{1}{n^2}

    The first sum is the harmonic series and diverges to infintiy and the 2nd sums to \frac{\pi^2}{6} is I remember correctly so the sum diverges to infinity.
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  3. #3
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    Quote Originally Posted by Pinkk View Post
    \sum\,\frac{n-1}{n^{2}}

    I'm having a brain fart, I don't know if this converges or not.
    limit comparison with the known divergent series \sum \frac{1}{n} ...

    \lim_{n \to \infty} \frac{\frac{n-1}{n^2}}{\frac{1}{n}}

    \lim_{n \to \infty} \frac{n-1}{n^2} \cdot \frac{n}{1} = 1

    series diverges
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  4. #4
    Senior Member Pinkk's Avatar
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    Hmm, that seems obvious enough, but since I'm in a real analysis course, I have to show this a bit more formally; it was never proven in my class that the sum/difference of a divergent and a convergent series is divergent.
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