Series (20 pts)

2007-03-01 5:36 am

a) Suppose that convergence of {(∞Σ n=1)an} can be established by ratio test. Prove that {(∞Σn=1)√(an)} converges. (10 marks)

b) If the convergence of {(∞Σ n=1)an} can be established by ratio test. Is it true that {(∞Σn=1)√(an)} will always converge？Give reason to justify your assertion. (10 marks)

更新1:

sorry，網友提醒，原來b打錯了，應是n√(an)，如下： b) If the convergence of {(∞Σ n=1)an} can be established by ratio test. Is it true that {(∞Σn=1)n√(an)} will always converge？Give reason to justify your assertion. (10 marks)

回答 (2)

魏王將張遼

2007-03-06 7:59 pm

✔ 最佳答案

(a)

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Maths/Sequencelim1.jpg

(b)

圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Maths/Sequencelim2.jpg

參考: My Maths knowledge

👍 0 👎 0

andycheungyatming

2007-03-03 1:33 am

(a)

Now, convergence of {(∞Σ n=1)a(n)} can be established by ratio test.
a(n+1) √a(n+1) a(n+1)
∃ R,---------- < R < 1. This implies ---------- = √----------- < √R < 1
 a(n) √a(n) a(n)
By ratio test, {(∞Σn=1)√(an)} converges.

[Ratio test can be proven by geometric series learnt in F.5, hence we need 0 < R < 1]

(b)

Simple Logics. [If (a) is true, then (b) can be answered easily]

👍 0 👎 0

收錄日期: 2021-04-26 11:51:06
原文連結 [永久失效]:
https://hk.answers.yahoo.com/question/index?qid=20070228000051KK04506

檢視 Wayback Machine 備份