Question $1.$

a$)$ Given the following infirite serios, where $y=2\mathrm{.}5.$

$c^y={\displaystyle \underset{k=0}{\overset{}{}}}\frac{y^x}{k!}=1+y+\frac{y^2}{2!}+$cdots

Using Matlab, write a for loop to sum the first twenty terms of the series and compute the relative error. Secondly, write a while loop that will determine how many terms must be summed to produce an approximation of $e^{2\mathrm{.}5}$ that is correct to $4$ significant figures. Use fprint f to output your results. Lastly, plot a well labelled figure of $e^U$ versus its indices. Submit both the code, the figure, and your output.

$[16]$