수열, 급수 및 균등 수렴
도입됨| English | 의미 |
|---|---|
| Cauchy sequence/ˈkɔːtʃi ˈsiːkwəns/ | 코시 수열 |
| uniform convergence/ˈjuːnɪfɔːm kənˈvɜːdʒəns/ | 일관적 수렴 |
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?
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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?
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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)^____.
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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.
다음 중 이 사례에 부합하는 답안은 무엇입니까?
코시, 단조 수렴 및 부분열 기준을 활용하다
Uniform convergence of differentiable functions guarantees convergence of their derivatives to the derivative of the limit.
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
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.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
난이도 3/5 · 200 코인 + 오늘 첫 완료 시 소액 보너스
The order follows the stated process; check each stage before the next.