Visual explanation of $\pi$ series definition Can you visually explain why the following is true:
$$
\frac{\pi}{4} = \sum\limits_{k=0}^\infty \frac{(-1)^k}{2k + 1} = \frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}\ldots\approx 78.5\%
$$
By visually I mean use a circle and square to help me understand rather than some calculus that I won't understand.
I have recently been very intrigued by $\pi /4$ as it seems to make geometry simpler. For example, the area of a circle is $\pi /4$ times the area of its circumscribed square, and the circle's perimeter is $\pi /4$ times the perimeter of its circumscribed square. See more here: Geometry with $\pi /4$

You should know that:
$$\dfrac{\pi}{4}=\tan^{-1}(1)$$
This means that:
$$\dfrac{\pi}{4}=\int_0^1 \dfrac{1}{1+x^2} \ dx$$
We will use the rule that $\dfrac{1}{1+x^2}=1-x^2+x^4-x^6+x^8\dots \ (-1\le x \le 1)$.
$$\dfrac{\pi}{4}=\int_0^1 1 -x^2+x^4-x^6+x^8\dots \ dx$$
We will solve the indefinite integral $\int 1 -x^2+x^4-x^6+x^8\dots \ dx$ first. This is basically power rule repeated an infinite number of times.
$$\int 1 -x^2+x^4-x^6+x^8\dots \ dx$$
$$= x - \dfrac{x^3}{3} + \dfrac{x^5}{5} - \dfrac{x^7}{7} + \dfrac{x^9}{9} \dots + C$$
Now we will evaluate the definite integral. We just need to use the Fundamental Theorem of Calculus to do this. The Fundamental Theorem of Calculus is this:
Suppose $G$ is an antiderivative of $f$. Then:
$$\int_a^b f(x) \ dx = G(b) - G(a)$$
To find the definite integral, we just have to plug in the value of $b$ (which is $1$) into the antiderivative (which is basically the answer to the integral) and evaluate it. Then we plug in $a$ to the antideravitave and evaluate it. Finally, we subtract the second value ($G(a)$) from the first value ($G(b)$).
$$\dfrac{\pi}{4}= \left(1 - \dfrac{1^3}{3}+\dfrac{1^5}{5}-\dfrac{1^7}{7}+\dfrac{1^9}{9}\dots + C \right) - \left(0 - \dfrac{0^3}{3}+\dfrac{0^5}{5}-\dfrac{0^7}{7}+\dfrac{0^9}{9}\dots + C \right)$$
$$\dfrac{\pi}{4}= 1 - \dfrac{1^3}{3}+\dfrac{1^5}{5}-\dfrac{1^7}{7}+\dfrac{1^9}{9}\dots + C - C$$
$$\displaystyle \boxed{\dfrac{\pi}{4}= 1 - \dfrac{1}{3}+\dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{9}\dots}$$
