Re: [積分] 一題不定積分 一題級數歛散

看板trans_math作者 (TMC)時間14年前 (2011/04/05 21:43), 編輯推噓9(9推 0噓 24→)
留言33則, 5人參與, 最新討論串2/2 (看更多)
※ 引述《kkgfdsaa (Jared)》之銘言： : 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 -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 163.22.18.57
推
04/05 22:24, , 1F
感謝 請問第二題呢?
04/05 22:24, 1F
推
04/05 22:51, , 2F
我忘了n^(1/n) 遞減到 1 怎麼證了= =
04/05 22:51, 2F
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04/05 22:51, , 3F
(只有前幾項會跳動 之後就一路遞減)
04/05 22:51, 3F
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04/05 22:51, , 4F
如果有這個當前提的話 n^(1/n)-1 就↓0
04/05 22:51, 4F
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04/05 22:53, , 5F
Consider An=n^(1/n)-1 ,n>=N0(確保遞減)
04/05 22:53, 5F
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04/05 22:53, , 6F
An+1-An=(n+1)^(1/n+1) - n^(1/n) < 0
04/05 22:53, 6F
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04/05 22:54, , 7F
注意到An>0 for all n , 除An 不變號
04/05 22:54, 7F
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04/05 22:55, , 8F
又因為An+1 An都>0 所以可直接掛絕對值
04/05 22:55, 8F
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04/05 22:55, , 9F
符合Ratio test的需求 收斂
04/05 22:55, 9F
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04/05 22:56, , 10F
我用Wolfram跑 收斂值是10^8左右XD
04/05 22:56, 10F
推
04/05 23:31, , 11F
應該可以不用Ratio test 反正monotonic
04/05 23:31, 11F
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04/05 23:31, , 12F
又bounded
04/05 23:31, 12F
推
04/05 23:33, , 13F
應該可以直接取極限 發現他收斂到1
04/05 23:33, 13F
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04/05 23:33, , 14F
在利用Z大說的遞減
04/05 23:33, 14F
推
04/06 09:02, , 15F
a_n = n^(1/n) -1. b_n = (ln n)/n
04/06 09:02, 15F
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04/06 09:04, , 16F
n^1/n=t => a_n/b_n = (t-1)/lnt
04/06 09:04, 16F
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04/06 09:05, , 17F
-> 1. So, Σ a_n diverges by limit
04/06 09:05, 17F
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04/06 09:05, , 18F
comparison test with Σ b_n diverges.
04/06 09:05, 18F
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04/06 09:06, , 19F
By the way, we can choose b_n = 1/n.
04/06 09:06, 19F
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04/06 09:06, , 20F
But it needs some more computations.
04/06 09:06, 20F
推
04/06 09:12, , 21F
For n^1/n ↓ by observing (1+1/n)^n
04/06 09:12, 21F
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04/06 09:13, , 22F
<= 3 <= n. ( (n+1)^n <= n^(n+1) )
04/06 09:13, 22F
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04/06 09:14, , 23F
But I dont know how to use Ratio test.
04/06 09:14, 23F
推
04/06 10:51, , 24F
回math大，(t-1)/lnt，應該不會收到1
04/06 10:51, 24F
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04/06 10:52, , 25F
所以comparison test 好像沒法說明是收斂
04/06 10:52, 25F
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04/06 10:53, , 26F
感謝z大，但ratio test 我還是test不出來
04/06 10:53, 26F
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04/06 10:54, , 27F
我知道n^(1/n)用自然對數形式就可知道是1
04/06 10:54, 27F
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04/06 10:55, , 28F
用nth test 知道是發散
04/06 10:55, 28F
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04/06 10:57, , 29F
之後用direct comparison 還是無法證出
04/06 10:57, 29F
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04/06 10:58, , 30F
所以還請各位高手幫忙一下
04/06 10:58, 30F
推
04/06 19:33, , 31F
(t-1)/lnt -> 1 (這是對的)
04/06 19:33, 31F
推
04/06 21:28, , 32F
yeah, as t→1
04/06 21:28, 32F
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04/07 17:04, , 33F
感謝各位
04/07 17:04, 33F
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文章代碼(AID): #1DcnoSAF (trans_math)