The limit definition: (1 + 1/n)ⁿ → e
Table showing (1+1/n)^n converging to e
| n | (1 + 1/n)ⁿ | Abstdan zu e |
|---|---|---|
| 1 | 2,000000 | 0,71828 |
| 10 | 2,593742 | 0,12454 |
| 100 | 2,704814 | 0,01347 |
| 1 000 | 2,716924 | 0,00136 |
| 1 000 000 | 2,718281 | 0,0000014 |
| ∞ | 2,71828… | 0 |
e^x: the only function that is its own derivative
At x=1, the height of the curve is e ≈ 2.718 and the slope of the tangent is also e. No other base b^x has this property.
Compound interest converges to e as compounding increases
Starting with $1 at 100% annual interest: compounding monthly gives $2.613, daily $2.714, every second $2.718. The limit as n→∞ is exactly e.
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Berapa nilai e?
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Generate the digits of Euler's Number e
e has no final digit
Euler's Number e is irrational. Its decimal expansion never ends and never repeats. The digits shown below are verified against the taylor series.
e = 1 + 1/1! + 1/2! + 1/3! +...
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