Problem $6\mathrm{.}1.(20$pts$)$ We have three ways to define the Euler number $e,$ as follow.

$($a$)$ The number $e$ is the number in the domain of the natural logarithm that

satisfiesln $e={\displaystyle \underset{1}{\overset{e}{}}}\frac{1}{t}dt=1$

$($b$)e=\underset{0}{lim}(1+{)}^{\frac{1}{}}=\underset{n}{lim}(1+\frac{1}{n}{)}^n$

$($c$)e={\displaystyle \underset{n=0}{\overset{}{}}}\frac{1}{n!}$

Is it possible to write codes to compute the number $e$ based on the above definitions?

Which definition provides the most efficient approach to approximate the number $e?$

In your answer, you should make sure the following:

$($a$)$ You need to submit the codes and the corresponding result.

$($b$)$ Your code for each should be based on an iterative algorithm. That is$,$ your

code should consist of a loop procedure and usually it needs the result of $n-$

$1$ or the result less than or equal to $n-1$ to obtain the result of $n.$

$($c$)$ You may include the part to tell which code provides the most efficient

approximation, in terms of the number of iterations or time.

$($d$)$ In each of the iteration, you may need to specify the stopping criteria such as

when you believe the precision is good enough and you should terminate the

loop.