Definition:Euler's Number/Limit of Sequence

Definition

The sequence $\sequence {x_n}$ defined as $x_n = \paren {1 + \dfrac 1 n}^n$ converges to a limit as $n$ increases without bound.

That limit is called Euler's Number and is denoted $e$.

Decimal Expansion

The decimal expansion of Euler's number $e$ starts:

$2 \cdotp 71828 \, 18284 \, 59045 \, 23536 \, 02874 \, 71352 \, 66249 \, 77572 \, 47093 \, 69995 \ldots$

Also see

• Equivalence of Definitions of Euler's Number

Source of Name

This entry was named for Leonhard Paul Euler.

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 1$: Special Constants: $1.2$
• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: $\S 1.2$: Real Sequences
• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $2 \cdotp 718 \, 281 \, 828 \, 459 \, 045 \, 235 \, 360 \, 287 \, 471 \, 352 \, 662 \, 497 \, 757 \, 247 \, 093 \, 699 \ldots$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $2 \cdotp 71828 \, 18284 \, 59045 \, 23536 \, 02874 \, 71352 \, 66249 \, 77572 \, 47093 \, 69995 \ldots$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): e
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): e
• 2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $5$: Eternal Triangles: The number $e$