Engineers and architects have needed to know the approximate value of $($the ratio of the circumference of a circle to its diameter$)$ for millenia, leading mathematicians to developed numerous approximation techniques over the ages. One approximation method is the Madhava$-$Leibniz formula$*,$

$={\displaystyle \underset{k=0}{\overset{}{}}}{a}_k=\frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+$dots

where

$a_k=\frac{4(-1{)}^k}{2k+1}$

Write a code to approximate $$ by calculating and adding the first $11$ terms $(k=0$ to $k=10)$ of this sequence. Store the result to a variable named piapprox.

"This formula is named after German mathematician Gottfried Leibniz, who, along with Scottish mathematician James Gregory, discovered it in the $1670$ s$.$ Had the Internet been aroupd at the time, they might have realized that Indian mathematician Madhava of Sangamagrama and his students were already using equivalent formulas two or three centuries eariier. While this formula can calculate $,$ it is very slow to converge. Much faster approximations have been developed in the centuries since.