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Sequences and their limits

What it means for aₙ to approach L, and the limits that define e and π.

A sequence converges to L when its terms eventually stay as close to L as you like. That “as you like” is the ε in every analysis proof. (1 + 1/n)ⁿ creeps up to e; n/(n + 1) creeps up to 1; (−1)ⁿ never settles. The graph shows the terms approaching the limit.

Fungerat exempel · limit of (1 + 1/n)^n as n -> oo

Calculus · Limits

Limit of (1 + 1/n)^n as n → oo

\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{n}

Steg för steg

  1. \lim_{n \to \infty^+-} \left(1 + \frac{1}{n}\right)^{n}

    Try direct substitution first.

  2. \

    As x grows without bound, compare the fastest-growing terms (or divide top and bottom by the highest power).

  3. = e

    Take the limit.

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e

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Symboler som används här

Knacka på alla symboler för den fullständiga definitionen, en bild, och vad varje bokstav i den betyder.

Hur man: Sequences and their limits

  1. Try direct substitution first.
  2. As x grows without bound, compare the fastest-growing terms (or divide top and bottom by the highest power).
  3. Take the limit.

Frågor folk frågar

What is the ε, δ definition actually saying?

That you can make the output as close to the limit as anyone demands (within ε) by keeping the input close enough (within δ). It replaces "approaches" with a challenge-and-response that can be checked.

Why does the harmonic series diverge when its terms go to zero?

Because the terms shrink too slowly: group them as 1/3 + 1/4 > 1/2, 1/5 + … + 1/8 > 1/2, and so on: infinitely many halves.

Mer information Real Analysis

Convergence of seriesImproper integralsTaylor approximation and errorFunctions of a complex variableContinuity and differentiability, rigorouslySequences of functions and uniform convergence