The exponential function is a miracle
The Maclaurin series for the exponential function converges for every complex number !! x !!: $$ 1 – x frac {x ^ 2} {2} – frac {x ^ 3} {6} frac {x ^ 4} {24} – ldots=e ^ {- x} $$
Say that !! x !! is any reasonably large number, such as 5. Then !! e ^ {- x} !! is close to zero, But the terms of the series arenotclose to zero. For !! x=5 !! we have: $$ 1 – 5 12 – 20. 83 26. 04 – 26. 04 21 – 15 5 9. 69 – 5. 38 ldots approx { Large 0 } $$
Somehow all these largish random numbers manage to cancel out almost completely. And the larger we make !! x !!, the more of these largish random numbers there are, the larger they are, and yet the more exactly they cancel out. For even as small an argument as !! x=20 !!, the series begins with 52 terms that vary between 1 and forty-three million, and these somehow cancel out almost entirely. The sum of these 52 numbers is !! – 0.4 !!.
In this graph, the red lines are the various partial sums (!! 1-x, 1-x frac {x ^ 2} 2, !! etc.) and the blue line is the total sum !! e ^ {- x} !!.
As you can see, each red line is a very bad approximation to the blue one, except within a rather narrow region. And yet somehow, it all works out in the end.
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