$2-[30$p$]$Harlan A$.$ Brothers and John A$.$ Knox discovered that as the value of $x$ gets larger, the value of the expression

$(\frac{2x+1}{2x-1}{)}^x$

gets closer and closer to e $.$ Write a program that evaluates this expression for $x=1,2,3,$ and so on until the absolute difference between the expression's value and the real value of $"e^{?}$ is less than $0\mathrm{.}000001.$ Display the value of $x$ that causes your loop to exit along with both the final approximation of $e$ and the real value of $"e".$ Show $6$ decimal places. $($Real e$=2\mathrm{.}718281)$

$[$Do not use math.h library. For example, If you need to calculate power of number, write your own function!$]$