I am new to this forum, but I am hoping I can ask for your help. I have recently been interested in finding and recording as many infinite series apporximating π as I can find. There is one in particular I have noted on an old copybook, but I have not seen it anywhere else. I will write it here hoping that someone might recognize it and tell me if it is in fact an approximating series of π and who came up with it.
$\pi=\displaystyle\sum_{i=1}^\infty \frac{4}{n\left(1+\left(\frac{i-\frac{1}{2}}{n}\right)^2\right)}$
Thank you in advance for your help.
You mean $$ \pi = \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{i=1}^n \frac{4}{1+\left( \frac{i-1/2}{n} \right)^2} \text{,} $$ which is a midpoint Riemann sum for the integral $$ \int_0^1 \; \frac{4}{1+x^2} \,\mathrm{d}x = \pi \text{.} $$ This evaluation follows a straightforward trigonometric substitution, $x = \tan \theta$: \begin{align*} \int_0^1 \; \frac{4}{1+x^2} \,\mathrm{d}x &= \int_0^{\pi/4} \; \frac{4 \sec^2 \theta} {1+\tan^2 \theta} \,\mathrm{d}\theta \\ &= \int_0^{\pi/4} \; 4 \,\mathrm{d}\theta \\ &= 4\left( \frac{\pi}{4} - 0 \right) \\ &= \pi \text{.} \end{align*}