Proving $\displaystyle \forall x\in\mathbb{Z}^+, \frac{1}{x}<\pi-4\sum_{n=1}^x\frac{x}{x^2+n^2}$?
I stumbled across this when messing with series that converge to Pi. The fact that $4\sum_{n=1}^x\frac{x}{x^2+n^2}$ converges to Pi as x approaches infinity follows from evaluating the integral on the right side of the below equation via a Riemann Sum and then simplifying:
$$\frac{\pi}{4} = \int_{0}^{1} \left(\frac{1}{1+x^2}\right)\mathrm dx$$
I noticed that the convergence rate seems to have a lower bound (not sure if that is the correct use of the term) of 1/x, but my attempts at a proof for this haven't really gone anywhere (made a feeble attempt at induction, which looked promising at first, but quickly got stuck). I am currently a HS student so I have a limited understanding of math, if anyone could show me the proof for this and dumb it down to my level (and even better the actual lowest possible bound of the expression) that would be super nice! Thanks so much!
P.S, first time posting here, normally I would post latex as images, but they seem to be disallowed on this sub. Hope inline latex is acceptable.
Posted Jan 28, 2019, 6:15 AM UTC
I've tried a few approaches without luck. If it's true the proof is probably a bit convoluted.
There are a few tricks which might help. You can change the inequality into an equivalent one, you can find an integral which gives a bound to the sum or you can try a different way of induction like proving that if it is true for k it is true for 2k and if it is true for 2k and 2k+2 it is true for 2k+1.
Even if you don't solve it I'd still suggest looking into methods you could use and try to apply them. You'll learn a lot by doing that.