TITLE: Where do Mathematicians Get Inspiration for Pi Formulas?
QUESTION [26 upvotes]: 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.

REPLY [4 votes]: One method, no doubt, is due to reasoning involving the classic definition of $pi$ (the ratio of the diameter to the circumference). For example the ratio of the diameter of a regular polygon to perimeter as the number of sides goes to infinity gives $\pi$. Starting with the square and doubling the number of sides of the polygon yields the sequence $$2\sqrt2$$ $$4\sqrt{2-\sqrt2}$$ $$8\sqrt{2-\sqrt{2+\sqrt2}}$$
$$16\sqrt{2-\sqrt{2+\sqrt{2+\sqrt2}}}$$ $$\dots$$
You could derive something similar for $\pi^2$ or $\pi^3$ using the areas of regular polygons, surface areas/volumes of convex regular polyhedrons, etc. 
Just speculating: one might get inspiration from physical phenomenon such as angular velocity vs linear velocity for an object traveling in a circle, angular momentum, torque, etc.