e is 2.718281828..... so as you can see, there is a pattern. So, I break it down into 2.7 + 0.01828 + 0.0000001828 .....
and then I use sum of infinity to get the fraction
Well, in mathematics theory , e cannot be in fraction. But I tried to use sum of infinity and proof it that it can be in fraction. Am I correct?
On
$e$ can be represented in a continued fraction. See the List of representations of $e$.
$e=2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{4+\cfrac{1}{1+\cfrac{1}{1+\ddots\,}}}}}}}=1+\cfrac{2}{1+\cfrac{1}{6+\cfrac{1}{10+\cfrac{1}{14+\cfrac{1}{18+\cfrac{1}{22+\cfrac{1}{26+\ddots\,}}}}}}}$
$e=2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{2}{3+\cfrac{3}{4+\cfrac{4}{5+\ddots}}}}}= 2+\cfrac{2}{2+\cfrac{3}{3+\cfrac{4}{4+\cfrac{5}{5+\cfrac{6}{6+\ddots\,}}}}}$
$e=2+\cfrac{1}{1+\cfrac{2}{5+\cfrac{1}{10+\cfrac{1}{14+\cfrac{1}{18+\ddots\,}}}}} =1+\cfrac{2}{1+\cfrac{1}{6+\cfrac{1}{10+\cfrac{1}{14+\cfrac{1}{18+\ddots\,}}}}}$
You have proved that
$$ e' = 2.718281828(1828)... $$
is rational. This is a special case of a more general statement that a number is rational if and only if its decimal expansion is repeating or terminating. However, $e'$ is not the same number as
$$ e = \lim_{n\to\infty}\left(1 + \frac{1}{n}\right)^n = 2.718281828\color{red}{45904}... . $$
In fact, it is a fairly well-known fact that $e$ is irrational (transcendental even). See for example this wikipedia article for a few proofs of this fact.
That said, your observation provides a convenient mnemonic for the first ten digits of $e$!