system
Given the following post, respond as if you're commenting on Reddit:
human
Why can we calculate (not measure) pi?

Understanding that pi represents the ratio of circumference to diameter of a circle, my question is regarding one (of several) methods to mathematically calculate the value of pi.

pi / 4 = 1 - 1/3 + 1/5 - 1/7 + ...

As you extend this equation it gets closer and closer to the value of pi. Admittedly it takes a ridiculous number of terms to extend it more than a few decimal places, but presumably it would continue to gain accuracy as you added/subtracted an infinite number of fractions.

My question: WHY? How does this equation relate to the characteristics of circles? Is something more profound at work? Enlighten me, please!
gpt
There is a way of approximating functions called a Taylor series expansion, where you add polynomial terms of higher and higher order to approximate the function. For example, the Taylor series of the cosine function is 1-x^2 /2 +x^4 /24 -x^6 /720... and if you calculate some values for x in radians, you'll find it's accurate if x isn't too big. You can figure out the terms with basic calculus.

From trigonometry, we know that the tangent of pi/4 radians (45 degrees) is 1. Thus, the inverse tangent of 1 is pi/4. The Taylor series for the inverse tangent function is x-x^3 /3 +x^5 /5 -x^7 /7 ... and if you substitute x=1, you're just left with that continued series for pi/4.