The following I saw as an exercise in a text on modular forms. I am lacking the understanding for why the following is true, but it is nevertheless astonishing:
Why is the numerical value of
$$x:=\sum_{n=1}^{\infty} \exp(-(n/10)^2)$$
so ridiculously close to the value of
$$y:=5\sqrt\pi-\frac12$$
while (and this is the truly surprising aspect) not being equal to it?
And by "ridiculously close" I mean that $x$ and $y$ agree up to the 427-th digit after the decimal point:
\begin{align} x=8. &3622692545275801364908374167057259139877472806\\ &1193564106903894926455642295516090687475328369\\ &2723327081134118121412853331180764328622113012\\ &6254685480139353423101884932655256142496258651\\ &4475413114466047689633981400087319507675739860\\ &2583500950926170092927234872474563201569608877\\ &6295310820270966625045319920380686673873757671\\ &6833994894682925918204397725582580869380029533\\ &6967158956664049274231240924510273274260978066\\ &257808237337\color{red}{62} \end{align}
They first disagree in the red digits.
Question: Can someone explain this in "simple" words? It does not have to be elementary, but I want to understand what machinery has to interact to arrive at this. And how would one even come up with such an example, or similar ones?