# Thread: Summation, possibly something cool?

1. ## Summation, possibly something cool?

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

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

$\displaystyle 1\leq \zeta(2) \leq 2$

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

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

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

Which is unrelated to the original problem, but it's something I did think about and I was wondering if it was correct ( $\displaystyle n \in \mathbb{R}$ ) 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.

2. ## Re: Summation, possibly something cool?

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

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

$\displaystyle 1\leq \zeta(2) \leq 2$

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

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

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

Which is unrelated to the original problem, but it's something I did think about and I was wondering if it was correct ( $\displaystyle n \in \mathbb{R}$ ) 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 $\displaystyle i=0$.

CB