Decide whether
inf
SUMMATION OF (a_n)
n=2
converges for
i) a_n = (3n + 4)/n^3
ii) a_n = (e^n)(n^2)/n!
iii) a_n = [cos(2n)]/(1 + n^2)
iv) a_n = 1/[n(ln(n))^(5/4)]
PS. plz imagine the "summation of" is the sign, the "inf" and "n=2" are above and below it respectively.
series convergence
2008-04-18 10:14 am
回答 (1)
2008-04-18 8:26 pm
✔ 最佳答案
(i)圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Apr08/Crazyseries1.jpg
(ii) Applying the ratio test:
圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Apr08/Crazyseries2.jpg
So the series converges since the absolute value of ratio test result is smaller than unity.
(iii) Through finding the upper bound of the series:
圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Apr08/Crazyseries3.jpg
(iv) Consider the sketch of a function f(x) = 1/[x(ln x)4/5] as follows:
圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Apr08/Crazyseries4.jpg
Then using the fact that the sum of the area of the rectangluar blocks from x = 2 to infinity is smaller than the area covered by the curve from 2 to infinity, and evaluation of improper integral, we have:
圖片參考：http://i117.photobucket.com/albums/o61/billy_hywung/Apr08/Crazyseries5.jpg
參考: My Maths knowledge
收錄日期: 2021-04-15 01:25:25
原文連結 [永久失效]:
https://hk.answers.yahoo.com/question/index?qid=20080418000051KK00260