The aim of this question is to approximate $$~~$3\mathrm{.}141592653589793$dots, using the series expansion

$={\displaystyle \underset{k=0}{\overset{}{}}}\frac{1}{{16}^k}(\frac{4}{8k+1}-\frac{2}{8k+4}-\frac{1}{8k+5}-\frac{1}{8k+6})$

Modify the above Matlab$/$Python script to approximate $$ using the series expansion $(1).$ What is the smallest number of terms necessary to obtain the same value as the one produced by

pi in Matlab $?$ Store this value in the variable npi. $($You may use trial and error to answer this question.$)$ Note that summation starts with $k=0$ so that the

number of terms is one larger then the index. example: n$=17$;

eapp $=0$;

for k $=0$:n

eapp $=$ eapp $+1/$factorial$($k$)$;

end

error $=$ abs$($eapp $-$ exp$(1))$