級數 (Series)
定義: 級數(series)
- 給定序列 {an}, 則(無窮)級數為 ∑n=1∞an.
- 此級數之部份和為 sn=∑k=1nan.
- 此級數收斂若 limn→∞sn=s, 常寫為 ∑n=1∞an=s.
- ∑n=1∞an converges ⇔ ∀ϵ>0 ∃n0∈N ∋ ∣∑k=nmak∣≤ϵ ∀m,n≥n0.
- Let m=n, then ∣an∣≤ϵ ∀n≥n0 ⇒ limn→∞an=0.
- 級數收斂的必要條件是數列必須收斂至0。(可解釋為如果在無窮多項時,數列之值仍不為0,則累加之值必須持續上升而發散。)
- E.g. limn→∞n1=0, but ∑n=1∞n1 diverges.
- Definition: 絕對收斂(absolute convergence)
- The series ∑n=1∞an is said to converge absolutely if the series ∑n=1∞∣an∣ converges (i.e. ∑n=1∞∣an∣≤∞)..
- 由定義可知 ∑∣an∣≤∞ ⇒ ∑an≤∞. (∵∣∑an∣≤∑∣an∣).
- 反之不成立,E.g. ∑n(−1)n.
- 對於每一個均為正值的數列,絕對收斂等價於一般的收斂 (an≥0⇔∣an∣=an)。
序列重排 (Series rearrangements)
- 序列重排是將原數列an的部份元素調換順序後形成的新數列an′,因此可以在兩數列中找到一個1-to-1 function。
- 如果sn=∑k=mnan且sn′=∑k=mnak′為兩數列的partial sum,可知在大部分的情況下sn≠sn′.
- Theorem: 若級數∑an收斂,但非絕對收斂,則重排後的級數∑an′可能會收斂到任意值,或是不收斂。
- Let ∑an be a series of real numbers, which converges but not absolute converges. Suppose that −∞≤a≤b≤∞, then there exists a rearrangement ∑an′ with the partial sum sn′ such that liminfn→∞sn′=a and limsupn→∞sn′=b.
- Theorem: 絕對收斂的級數,經過序列重排後,仍然收斂到相同值。
- If ∑an is a series of complex numbers which converges absolutely, then every rearrangement of ∑an converges and they all converges to the same sum.
級數審斂法 (Series test)
- Exponential number e=∑n=0∞n!1
- limn→∞(1+n1)n=e.
根值審斂法 (root test)
- Given ∑an, let limsupn→∞∣an∣1/n=a then
- If a<1, ∑an converges.
- If a>1, ∑an diverges.
- If a=1, the test gives no information.
- 實際使用此法時,取a=limn→∞∣an∣1/n.
比例審斂法 (Ratio test)
- Given ∑an, then
- if limsupn→∞∣anan+1∣<1, ∑an converges.
- If ∣anan+1∣≥1 for n≥n0, ∑an diverges.
- Ratio test比root test更直覺且容易計算,但root test的用途更廣。
- Theorem: For any sequence {cn} of positive numbers
- liminfn→∞cncn+1≤liminfn→∞cn1/n.
- limsupn→∞cn1/n≤limsupn→∞cncn+1.
- 此定理說明了若比例審斂法收斂 ⇒ 根值審斂法也收斂。