Definitions of e
- Limit: e = lim (1 + 1/n)ⁿ as n → ∞
- Series: e = Σ 1/k! = 1 + 1 + 1/2 + 1/6 + 1/24 + … (k from 0 to ∞)
- Logarithm: e is the unique number with ln e = 1
- Calculus: eˣ is the only exponential function equal to its own derivative, d/dx eˣ = eˣ
- Continuous compounding: $1 at 100% interest compounded continuously for one year grows to $e
Sources: NIST Digital Library of Mathematical Functions: DLMF §4.2 Logarithm, Exponential, Powers: Definitions; OpenStax: Calculus Volume 2 §6.3 Taylor and Maclaurin Series
The value of e to 50 decimal places
e = 2.71828182845904523536028747135266249775724709369995…
These digits are computed here from the series Σ 1/k! in exact integer arithmetic with guard digits, then truncated, and they agree with the NIST DLMF and OEIS A001113 values.
How the definitions converge
The series converges very quickly: each extra term adds roughly one more correct digit once k passes 10. The limit converges slowly: (1 + 1/n)ⁿ is off by about e/(2n).
| Method | Value | Error vs e |
|---|---|---|
| (1 + 1/n)ⁿ, n = 1 | 2.0000000000 | 7.18 × 10⁻¹ |
| (1 + 1/n)ⁿ, n = 10 | 2.5937424601 | 1.25 × 10⁻¹ |
| (1 + 1/n)ⁿ, n = 100 | 2.7048138294 | 1.35 × 10⁻² |
| (1 + 1/n)ⁿ, n = 1,000 | 2.7169239322 | 1.36 × 10⁻³ |
| (1 + 1/n)ⁿ, n = 10,000 | 2.7181459268 | 1.36 × 10⁻⁴ |
| (1 + 1/n)ⁿ, n = 1,000,000 | 2.7182804693 | 1.36 × 10⁻⁶ |
| Σ 1/i! for i = 0…2 | 2.5000000000 | 2.18 × 10⁻¹ |
| Σ 1/i! for i = 0…5 | 2.7166666667 | 1.62 × 10⁻³ |
| Σ 1/i! for i = 0…10 | 2.7182818011 | 2.73 × 10⁻⁸ |
| Σ 1/i! for i = 0…15 | 2.7182818285 | 5.02 × 10⁻¹⁴ |