Sequences, series and uniform convergence
| English | 中文 | Pinyin |
|---|---|---|
| Cauchy sequence/ˈkɔːtʃi ˈsiːkwəns/ | 柯西序列 | kē xī xù liè |
| uniform convergence/ˈjuːnɪfɔːm kənˈvɜːdʒəns/ | 一致收敛 | yí zhì shōu liǎn |
A decision before an answer
- Each curve x^n is continuous on [0,1], yet its pointwise limit has a jump. What assumption would prevent that jump?
- Your goal: Use Cauchy, monotone convergence and subsequence criteria.
Read the relationship
- A convergent real sequence is Cauchy, and every real Cauchy sequence converges because R is complete. In Q a Cauchy sequence can approach an irrational number and fail to converge within Q. A bounded monotone real sequence converges. A bounded sequence need not converge, but Bolzano–Weierstrass guarantees a convergent subsequence; (-1)^n has two different subsequential limits.
- Distinguish pointwise from uniform convergence of functions.
Which statement must hold for a bounded real sequence?
Bolzano–Weierstrass gives a convergent subsequence. The sequence (-1)^n rules out the other necessary claims.
Use the defining rule
- For a numerical series, absolute convergence implies convergence; conditional convergence does not allow arbitrary rearrangement without affecting the sum. The alternating harmonic series converges but its absolute-value series diverges. In a ratio test, a limit below 1 proves absolute convergence and one above 1 proves divergence; a limit equal to 1 is inconclusive, as both sum 1/n and sum 1/n² demonstrate.
- Check hypotheses before interchanging limits and integrals.
For f_n(x)=sin(nx)/n on R, convergence to zero is what type?
The supremum of the absolute value is 1/n, independent of x, which tends to zero.
Check the conditions
- Pointwise convergence chooses an index N separately for each x and tolerance. Uniform convergence chooses one N that works for every x in the domain. For real-valued functions, check the supremum of |f_n−f| over the whole domain. A continuous pointwise limit does not by itself prove uniform convergence. Domain endpoints and shrinking peaks often distinguish the two notions.
- Check hypotheses before interchanging limits and integrals.
On [0,1], f_n(x)=x^n tends to 0 for x<1 and to 1 at x=1. The limit is discontinuous, so convergence cannot be uniform; directly, the supremum error is 1, approached below 1. On [0,a] with 0≤a<1, the limit is zero and the supremum is a^n, which tends to zero, so convergence is uniform. Changing the domain changes the conclusion.
The supremum of |x^n| on [0,0.5] is (0.5)^____.
For positive integer n the function is increasing there, so the maximum is its endpoint value.
Apply the task format
- A uniform limit of continuous functions is continuous. On a closed bounded interval, uniform convergence of Riemann-integrable functions permits exchanging limit and integral. Exchanging derivatives needs extra hypotheses; uniform convergence of the functions alone is insufficient. The Weierstrass M-test establishes uniform absolute convergence of a series of functions if each term is bounded by M_n on the whole domain and sum M_n converges.
- Check hypotheses before interchanging limits and integrals.
Checking several fixed x values proves neither a supremum bound nor uniform convergence. The point producing the largest error may move with n.
Which answer fits this case?
Use Cauchy, monotone convergence and subsequence criteria
Uniform convergence of differentiable functions guarantees convergence of their derivatives to the derivative of the limit.
sin(nx)/n tends uniformly to zero, but its derivative cos(nx) equals 1 at x=0 for every n, whereas the limit function has derivative zero.
Keep the distinctions
- Cauchy sequence 柯西序列 — A sequence whose sufficiently late terms are arbitrarily close to one another.
- uniform convergence 一致收敛 — Convergence with one error threshold index valid at every point of the domain.
- Use Cauchy, monotone convergence and subsequence criteria.
- Distinguish pointwise from uniform convergence of functions.
- Check hypotheses before interchanging limits and integrals.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.