Infinite Series

kingwinner

Determine convergence for each of the following:
∞
∑  1 / [n (log n)^2]
n=2

∞
∑  1 / [n log n log(log n)]
n=2

[log=ln=natural log]

呢2題唔識做......
pls help!

Naozumi

第1題用integral test la, set f(x) = 1/x(ln x)^2

第2題 都係

kingwinner

第2題用到 Integral test 咩? 可以教我點用 ma?

Integral test: if f is continuous, decreasing, and POSITIVE on [2,∞), then
∞
∑ f(k) converges if and only if
k=2
∞
∫ f(x) dx converges.
2

f(x) = 1 / [x log x log(log x)]
f is not positive on [2,∞)
e.g. f(2)<0, f(2.1)<0

Naozumi

咁你可以由 [3, ∞) , 唔好咁死板啦

kingwinner

∞
∫ f(x) dx
k

k 既數值對 convergence/divergence 有冇影響?

∞
∫ f(x) dx converges
8
∞
∫ f(x) dx  diverges
5
有可能嗎?

[ 本帖最後由 kingwinner 於 10-3-10 12:41 PM 編輯 ]

Naozumi

k 既數值對 convergence/divergence 有冇影響?
當然冇!!

加/減 finite 咁多terms係對個series 的convergence冇影響 !!

kingwinner

∫ 1 / [x (ln x)^2] dx = -1/ln(x) + C

∞
∫ 1 / [x (ln x)^2] dx = 0 - (-1/ln2) = 1/ln2
2
所以第一個 sum converge, o岩 ma?




∫1 / [x ln x ln(ln x)] = ln(ln(ln x)) + C

∞
∫1 / [x ln x ln(ln x)] = ∞ (diverges)
3
所以第2個 sum diverge?



好耐冇掂過 infinite series 了, 唔知做得o岩唔o岩?

Naozumi

第一題都要用3開始, ln 2 < 0
第2題加少少explanation
∞
∑  1 / [n log n log(log n)] diverges
n=3

所以
∞
∑  1 / [n log n log(log n)] 都係diverge
n=2

kingwinner

但第一題係 (ln x)^2 wor, (ln 2)^2 >0
f(x) = 1 / [x (ln x)^2]
f(x) >0 for all x>0

所以用 2 開始都 ok?

Naozumi

係我發神經, ln 2 > 0,所以唔洗由3開始

kingwinner

∞
∑  [n^(2n)] / (2n)!
n=1

呢題係咪用 ratio test?
Let a(n) = [n^(2n)] / (2n)!

a(n+1)/a(n) = (n+1)^(2n+2) / [(2n+2)(2n+1) n^(2n)]
但跟住點做?

thx!

Naozumi

1.   lim [(n+1)/n]^(2n) = e^2
2.   lim (n+1)^2/ [(2n+2)(2n+1)] = 1/4
3.    e> 2 , so????