Number e
The number e = 2,7182… is the sum of inverse factorials
Explanation
Only for the function f (x) = ex applies
and for x = 0 it has the value 1. Can we develop an infinite series for this based on this data? Let's give it a try. During differentiation, a power decreases by 1. We start with the series
and get
That's not right. The series now starts with a 1 (we had forgotten about that for a moment) and the exponents have yet to be captured. So a second try
and now we see
That's still not right either, because the fractions have all disappeared. We must think of something to keep them. A third attempt with
and see
It seems to be correct. And that actually makes sense. But we check it by differentiating one more time
That goes well, and also for x = 0 the value always remains 1. You can write this with factorials as
Now we can calculate the number e, because for x = 1 we get
In this, the successive terms are
1/0! = 1,0000
1/1! = 1,0000
1/2! = 0,5000
1/3! = 0,1667
1/4! = 0,0417
1/5! = 0,0083
1/6! = 0,0014
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e = 2,718…
The answer is accurate to 3 places.