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Math Help - Summation, possibly something cool?

  1. #1
    Member integral's Avatar
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    Summation, possibly something cool?

    I've been trying to solve \sum^{\infty}_{i=0}\frac{1}{i^2} and I have come up with the inequality

    \int^{\infty}_{1}\frac{1}{x^2}dx \leq \sum^{\infty}_{i=0}\frac{1}{i^2} \leq \int^{\infty}_{1}\frac{1}{x^2}dx +1
    or

    1\leq \zeta(2) \leq 2

    I've never studied proofs before, at all but it does seem logical, just by looking at the problem that

    \int^{\infty}_{1}\frac{1}{x^n}dx \leq \sum^{\infty}_{i=0}\frac{1}{i^n} \leq \int^{\infty}_{1}\frac{1}{x^n}dx +1
    or

    \int^{\infty}_{1}\frac{1}{x^n}dx \leq \zeta(n) \leq \int^{\infty}_{1}\frac{1}{x^n}dx +1

    Which is unrelated to the original problem, but it's something I did think about and I was wondering if it was correct ( n \in \mathbb{R} ) or if you could give a proof

    If you do give me a proof ( or disproof) of that, it would be nice if it was in terms I could understand, I have yet to officially take anything past integration by parts in calculus two.
    Last edited by integral; September 18th 2011 at 03:51 PM.
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  2. #2
    Grand Panjandrum
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    Re: Summation, possibly something cool?

    Quote Originally Posted by integral View Post
    I've been trying to solve \sum^{\infty}_{i=0}\frac{1}{i^2} and I have come up with the inequality

    \int^{\infty}_{1}\frac{1}{x^2}dx \leq \sum^{\infty}_{i=0}\frac{1}{i^2} \leq \int^{\infty}_{1}\frac{1}{x^2}dx +1
    or

    1\leq \zeta(2) \leq 2

    I've never studied proofs before, at all but it does seem logical, just by looking at the problem that

    \int^{\infty}_{1}\frac{1}{x^n}dx \leq \sum^{\infty}_{i=0}\frac{1}{i^n} \leq \int^{\infty}_{1}\frac{1}{x^n}dx +1
    or

    \int^{\infty}_{1}\frac{1}{x^n}dx \leq \zeta(n) \leq \int^{\infty}_{1}\frac{1}{x^n}dx +1

    Which is unrelated to the original problem, but it's something I did think about and I was wondering if it was correct ( n \in \mathbb{R} ) or if you could give a proof

    If you do give me a proof ( or disproof) of that, it would be nice if it was in terms I could understand, I have yet to officially take anything past integration by parts in calculus two.
    Fix your sum the first term cannot be i=0.

    CB
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