maths.free › Real Analysis › Sequences and their limits

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.

Delovni primer · 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}

Korak po koraku

  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.

Odkrij odgovor
e

Zdaj pa ti. Izberite problem, ali tipkajte ali narišite svojo. Vsak korak, sliko, odgovor skriti, dokler ne vprašate.

Obdrži svoje delo.

Brezplačen račun dodaja opombe o vsaki lekciji, zapis o tem, kar ste končali, vaše rešene težave na enem mestu, in tutor lahko vprašate o tej strani. Sama matematika je odprta za vse, prijavljena ali ne.

Registriraj se Prijava

Simboli, ki se uporabljajo tukaj

Tapnite vsak simbol za popolno opredelitev, sliko in kaj pomeni vsaka črka v njej.

Kako: 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.

Vprašanja, ki jih ljudje sprašujejo

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.

Več v Real Analysis

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