Euler's Number Math Example 1

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Example 1

easy
Evaluate lim⁡n→∞(1+1n)n\lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n and state what this limit defines.

Solution

  1. 1
    Substitute increasing values of nn: for n=10n=10, (1+110)10≈2.5937\left(1+\frac{1}{10}\right)^{10} \approx 2.5937; for n=100n=100, ≈2.7048\approx 2.7048; for n=1000n=1000, ≈2.7169\approx 2.7169.
  2. 2
    Observe that as n→∞n \to \infty the expression approaches a fixed value that does not grow without bound.
  3. 3
    By definition, lim⁡n→∞(1+1n)n=e≈2.71828…\lim_{n \to \infty}\left(1+\frac{1}{n}\right)^n = e \approx 2.71828\ldots, Euler's number.

Answer

e≈2.71828e \approx 2.71828
Euler's number ee is defined as this limit. It is irrational and transcendental, appearing naturally in continuous growth, calculus, and complex analysis.

About Euler's Number

Euler's number e≈2.71828e \approx 2.71828 is the unique base for which the exponential function exe^x is its own derivative — the natural base for growth and decay.

Learn more about Euler's Number →

More Euler's Number Examples

Example 2 hard

Show that the derivative of [formula] is itself, i.e., [formula], using the limit definition of the

Example 3 easy

Using a calculator, compute [formula] to four decimal places. Then determine whether [formula].

Example 4 medium

A bank offers continuous compounding at an annual rate of [formula]. Using [formula], find how much

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