[單變] 級數審斂一題

看板trans_math作者 (都射了你還吹)時間15年前 (2009/05/11 01:09), 編輯推噓3(3推 0噓 16→)
留言19則, 4人參與, 最新討論串1/1
sigma(n從1到無限大) [ (-1)^n-1 ] ( n^(1/n) -1 ) 判斷他是發散 絕對收斂 或條件收斂 -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 118.168.93.77
推
05/11 01:16, , 1F
發散
05/11 01:16, 1F
→
05/11 01:17, , 2F
用alternating series test
05/11 01:17, 2F
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05/11 01:19, , 3F
( n^(1/n) -1 )這一項用羅必達趨近1
05/11 01:19, 3F
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05/11 01:21, , 4F
可知第n項沒有趨近0所以發散
05/11 01:21, 4F
推
05/11 01:25, , 5F
哪有.. n^(1/n)趨近於1呀
05/11 01:25, 5F
推
05/11 01:25, , 6F
少看到一個1
05/11 01:25, 6F
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05/11 01:26, , 7F
所以1-1=0
05/11 01:26, 7F
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05/11 01:26, , 8F
alternating series test 收斂
05/11 01:26, 8F
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05/11 19:33, , 9F
第n項趨近於0並不一定是收斂吧....
05/11 19:33, 9F
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05/11 19:42, , 10F
那是第n項測試 現在用的是交錯級數測試
05/11 19:42, 10F
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05/11 19:43, , 11F
只要符合遞減 影響正負號那項不看其他項
05/11 19:43, 11F
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05/11 19:44, , 12F
第n項趨近於0就收斂了
05/11 19:44, 12F
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05/11 21:47, , 13F
那請問是絕對收斂還是條件收斂？
05/11 21:47, 13F
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05/11 21:47, , 14F
還要看看他的正項級數是否收斂吧
05/11 21:47, 14F
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05/12 00:14, , 15F
可證 n^{1/n}-1↓0, 故由交錯級數收斂定理
05/12 00:14, 15F
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05/12 00:14, , 16F
知原級數收斂.
05/12 00:14, 16F
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05/12 00:16, , 17F
又:(n^{1/n}-1)/(1/n)→∞,故Σ(n^{1/n}-1)
05/12 00:16, 17F
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05/12 00:17, , 18F
發散. 故原級數是條件收斂.
05/12 00:17, 18F
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05/12 00:52, , 19F
哦，多謝了
05/12 00:52, 19F
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