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.
Voorbeeld van werk · limit of (1 + 1/n)^n as n -> oo
Limit of (1 + 1/n)^n as n → oo
Stap voor stap
- \lim_{n \to \infty^+-} \left(1 + \frac{1}{n}\right)^{n}
Try direct substitution first.
- \
As x grows without bound, compare the fastest-growing terms (or divide top and bottom by the highest power).
- = e
Take the limit.
Onthul het antwoord
Nu jij. Kies een probleem, type of teken je eigen. Elke stap, een foto, het antwoord verborgen totdat je het vraagt.
Hier gebruikte symbolen
Tik op elk symbool voor de volledige definitie, een afbeelding en wat elke letter erin betekent.
Hoe: Sequences and their limits
- Try direct substitution first.
- As x grows without bound, compare the fastest-growing terms (or divide top and bottom by the highest power).
- Take the limit.
Vragen die mensen stellen
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.
Meer in Real Analysis
Convergence of seriesImproper integralsTaylor approximation and errorFunctions of a complex variableContinuity and differentiability, rigorouslySequences of functions and uniform convergence