Using Sagemath's programing language in a jupyter lab notebook $,$ solve :

a$)$ Create a list, named CL$1$ whose entries are the values $\frac{1}{n+1}(\frac{2n}{n}),$ where $0n10$

b$)$ Create a list, named CL$2$ whose entries are the values $(\frac{2n}{n})-(\frac{2n}{n+1}),$ where $0n10$

c$)$ Create list, named CL$3$ whose entries are the values $\frac{1}{2}{\displaystyle \underset{0}{\overset{4}{}}}{x}^n\sqrt[\phantom{2}]{\frac{4-x}{x}}dx,$ where $0n10.$

d$)$ Create list, named CL$4$ whose entries are the values $\frac{2}{}{4}^n{\displaystyle \underset{-1}{\overset{1}{}}}{t}^{2n}\sqrt[\phantom{2}]{1-t^2}dt,$ where $0n10.$

If done correctly, you will see that all of these lists are the same! In fact, we have been computing the "Catalan numbers", $C_n=\frac{1}{n+1}(\frac{2n}{n}).$

e$)$ Compute the sum ${\displaystyle \underset{n=0}{\overset{}{}}}\frac{C_n}{4^n}.$

Remark: If we wanted to prove that the output of the sum in part $($e$)$ is correct, we would use the description of the Catalan numbers give in part $($d$).$ Can you figure out how?