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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.

작업 예제: 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}

단계별로

  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.

답을 드러내세요
e

Symbols used here

\lim_{x \to a} f(x)
limit
The value f(x) approaches as x approaches a.
\infty
infinity
Not a number: "grows without bound" in limits and intervals.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\sup,\ \inf
supremum, infimum
Least upper bound, greatest lower bound.
|x|
absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.
e
Euler's number
2.71828…, the base whose exponential is its own derivative.
\sum_{k=1}^{n} a_k
summation
Add a_k for k = 1 up to n.
\prod_{k=1}^{n} a_k
product
Multiply a_k for k = 1 up to n.
\int f(x)\,dx,\ \int_a^b
integral
Antiderivative (indefinite) or signed area from a to b (definite).
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.
\forall,\ \exists
for all, there exists
Quantifiers: every x; at least one x.
\blacksquare\ \text{or}\ \square
end of proof (halmos)
Marks the point where the statement has been established.
\varepsilon,\ \delta
epsilon, delta
Small positive tolerances in the definition of a limit.
B(x, r),\ \overline{A},\ \partial A
open ball, closure, boundary
Points within r of x; A plus its limit points; the edge of A.

How to: 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.

Questions people ask

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.

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