Euler Reborn

Here is how you find successive convergents in $Mathematica$

Convergents[E,15]

returns 15 convergents:
$2,3,\frac{8}{3},\frac{11}{4},\frac{19}{7},\frac{87}{32},\frac{106}{39},\frac{193}{71},\frac{1264}{465},\frac{1457}{536},\frac{2721}{1001},\frac{23225}{8544},\frac{25946}{9545},\frac{49171}{18089},\frac{517656}{190435}$

here is the program that answers the question

Numerator@#+Denominator@#&@Last@Select[Convergents[E,20],Numerator@#<10000&]

returns $\boxed{3722}$ .

A key selection of successive convergents of the continued fraction expansion of $e$ are $\tfrac{1457}{536}$ , $\tfrac{2721}{1001}$ and $\tfrac{23225}{8544}$ . The last one gives too large a value of $m$ , so we want $2721 + 1001 = \boxed{3722}$ .

Again, no easy way to solve this program. Below is a Python 3.4 program (it literally tries all possible values of $m$ and $n$ ) which can be used to solve the problem efficiently. If we look at all of the contents in the list of fractions, we find the closest value possible is $2.7182817182817183$ , with the smallest possible values $m = 2721$ and $n = 1001$ . Note: Not all of the contents in the list are shown.