作者TsungMingC (TMC)
看板trans_math
标题Re: [积分] 一题不定积分 一题级数歛散
时间Tue Apr 5 21:43:53 2011
※ 引述《kkgfdsaa (Jared)》之铭言：
: 1.∫(e^x)/x dx = ?
1 2 3
∫ --- ( 1+ x+ x /2! + x /3! + ... )dx
x
1 2
= ∫ [--- + 1 + x/2! + x /3! + ... )dx
x
2 3
= ln|x| + x + x /2*2! + x /3*3! + ...
∞ n
= ln|x| + Σ x / n*n!
n=1
: ∞
: 2. Σ [n^(1/n)-1] is converges or diverges ?
: n=1
--
※ 发信站: 批踢踢实业坊(ptt.cc)
◆ From: 163.22.18.57
1F：推 kkgfdsaa:感谢 请问第二题呢? 59.114.197.132 04/05 22:24
2F：推 znmkhxrw:我忘了n^(1/n) 递减到 1 怎麽证了= =111.251.226.167 04/05 22:51
3F：→ znmkhxrw:(只有前几项会跳动 之後就一路递减)111.251.226.167 04/05 22:51
4F：→ znmkhxrw:如果有这个当前提的话 n^(1/n)-1 就↓0111.251.226.167 04/05 22:51
5F：→ znmkhxrw:Consider An=n^(1/n)-1 ,n>=N0(确保递减)111.251.226.167 04/05 22:53
6F：→ znmkhxrw:An+1-An=(n+1)^(1/n+1) - n^(1/n) < 0111.251.226.167 04/05 22:53
7F：→ znmkhxrw:注意到An>0 for all n , 除An 不变号111.251.226.167 04/05 22:54
8F：→ znmkhxrw:又因为An+1 An都>0 所以可直接挂绝对值111.251.226.167 04/05 22:55
9F：→ znmkhxrw:符合Ratio test的需求 收敛111.251.226.167 04/05 22:55
10F：→ znmkhxrw:我用Wolfram跑 收敛值是10^8左右XD111.251.226.167 04/05 22:56
11F：推 steve1012:应该可以不用Ratio test 反正monotonic 114.34.202.142 04/05 23:31
12F：→ steve1012:又bounded 114.34.202.142 04/05 23:31
13F：推 steve1012:应该可以直接取极限 发现他收敛到1 114.34.202.142 04/05 23:33
14F：→ steve1012:在利用Z大说的递减 114.34.202.142 04/05 23:33
15F：推 math1209:a_n = n^(1/n) -1. b_n = (ln n)/n 140.113.25.169 04/06 09:02
16F：→ math1209:n^1/n=t => a_n/b_n = (t-1)/lnt 140.113.25.169 04/06 09:04
17F：→ math1209:-> 1. So, Σ a_n diverges by limit 140.113.25.169 04/06 09:05
18F：→ math1209:comparison test with Σ b_n diverges. 140.113.25.169 04/06 09:05
19F：→ math1209:By the way, we can choose b_n = 1/n. 140.113.25.169 04/06 09:06
20F：→ math1209:But it needs some more computations. 140.113.25.169 04/06 09:06
21F：推 math1209:For n^1/n ↓ by observing (1+1/n)^n 140.113.25.169 04/06 09:12
22F：→ math1209:<= 3 <= n. ( (n+1)^n <= n^(n+1) ) 140.113.25.169 04/06 09:13
23F：→ math1209:But I dont know how to use Ratio test. 140.113.25.169 04/06 09:14
24F：推 kkgfdsaa:回math大，(t-1)/lnt，应该不会收到1 59.114.204.101 04/06 10:51
25F：→ kkgfdsaa:所以comparison test 好像没法说明是收敛 59.114.204.101 04/06 10:52
26F：→ kkgfdsaa:感谢z大，但ratio test 我还是test不出来 59.114.204.101 04/06 10:53
27F：→ kkgfdsaa:我知道n^(1/n)用自然对数形式就可知道是1 59.114.204.101 04/06 10:54
28F：→ kkgfdsaa:用nth test 知道是发散 59.114.204.101 04/06 10:55
29F：→ kkgfdsaa:之後用direct comparison 还是无法证出 59.114.204.101 04/06 10:57
30F：→ kkgfdsaa:所以还请各位高手帮忙一下 59.114.204.101 04/06 10:58
31F：推 math1209:(t-1)/lnt -> 1 (这是对的) 140.113.25.169 04/06 19:33
32F：推 alasa15:yeah, as t→1 111.249.0.47 04/06 21:28
33F：→ kkgfdsaa:感谢各位 123.240.24.194 04/07 17:04