Question:
- Where do people get their inspirations for $\pi$ formulas?
- Where do they begin with these ideas?
Equations such as$$\dfrac 2\pi=1-5\left(\dfrac 12\right)^3+9\left(\dfrac {1\times3}{2\times4}\right)^3-13\left(\dfrac {1\times3\times5}{2\times4\times6}\right)^3+\&\text{c}.\tag{1}$$$$\dfrac {2\sqrt2}{\sqrt{\pi}\Gamma^2\left(\frac 34\right)}=1+9\left(\dfrac 14\right)^4+17\left(\dfrac {1\times5}{4\times8}\right)^4+25\left(\dfrac {1\times5\times9}{4\times8\times12}\right)^4+\&\text{c}.\tag{2}$$$$\dfrac \pi4=\sum\limits_{k=1}^\infty\dfrac {(-1)^{k+1}}{2k-1}=1-\dfrac 13+\dfrac 15-\&\text{c}.\tag{3}$$Have always confused me as to where Mathematicians always get their inspirations or ideas for these kinds of identities.
The first one was found by G. Bauer in $1859$ (something I still want to know how to prove. I've found this recently asked question still open for proofs), the second was found by Ramanujan. And has a relation with Hypergeometrical series.
I'm wondering whether people see $\pi$ in other formulas, such as$$\sum\limits_{k=1}^{\infty}\dfrac 1{k^2}=\dfrac {\pi^2}6\implies\pi=\sqrt{\sum\limits_{k=1}^\infty\dfrac 6{k^2}}\tag{4}$$And isolate $\pi$, or if something new comes up and they investigate it?
For example, I'm wondering if it's possible to manipulate the expansion of $\ln m$
$$\ln m=2\left\{\dfrac {m-1}{m+1}+\dfrac 13\left(\dfrac {m-1}{m+1}\right)^3+\dfrac 15\left(\dfrac {m-1}{m+1}\right)^5+\&\text{c}.\right\}\tag{5}$$
To get a $\pi$ formula. Or the series$$\sum\limits_{k=1}^{\infty}\dfrac 1{k^p}=\dfrac {\pi^p}n\tag{6}$$Which converges faster and faster as $p$ gets larger and larger.