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Power series for the number e
The number e = 2.7182…can be expressed as the power series
Explanation
We start with the function
Substitution in the definition of the derivative gives
Evaluation shows
For x = 0 you get
This is a constant, as it does't contain any x variable. The value only depends on a, the base. So you can write
Now we want to know for what value of a the function f ′(0) = 1, then in this case f ′(x) = ax, and the function is its own derivative. So we calculate
Multiplication by h
Reshuffle
Simplification
Set , and because h → 0 this leads to n → ∞, and so
We use the binomium development, and obtain
This was a limit, so we write
If we apply l'Hôpital's rule several times on the different terms, we get
The subsequent terms yield
1 = 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 result is correct for 3 digits.