Consider the Taylor series representation of the exponential function

$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+$cdots

$($a$)$ For what values of $x$ does this series converge?

$($b$)$ Use Taylor's remainder theorem to find an expression relating the number of terms $N$ retained in the Taylor series to the relative error $lon$ of its approximation to $e^x.$

$($c$)$ Make a plot of $lon$ versus $N$ for $x=10,20,40.$ Use this plot to determine how many terms $N$ we must take to achieve a relative error of $1\%.$

$($d$)$ Explain why it is hard to compute an accurate approximation to $e^x$ for $x-1$ using floating point arithmetic.

$[$Hint: use your previous results and the fact the Taylor series is alternating for