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)

2007-03-06 7:59 pm
參考: My Maths knowledge
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]


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