作者wind0630 (小松鼠)
看板Math
標題[微積] 一題高微有地方不懂
時間Wed Jun 8 15:55:49 2011
Let f(x)=Σn^(-2)sinnx.Show that f is a continuous function on R
(1~無限大)
and the ∫f(x)dx=Σn^(-3) + 2Σn^(-3).
(0~拍/2) 1.3.5.. n=2.4.6...
<sol>:
The series defining f converges absolutely and uniformly on R
by the M-test with Mn=n^(-2).
Hence,f is continuous,and termwise integration is permissible
∫f(x)dx = Σn^(-2)∫sinnx dx = Σ(-n^(-3)cosnx)│0,拍/2
(0~拍/2) (1~無限大) (0~拍/2) (1~無限大)
Now,cos1/2n拍 is 0 when n is odd and (-1)^(n/2) when n is even.
Hence the nth term of the last series is n^(-3) when n is odd,
2n^(-3) when n=2.6.10...and 0 when n =4.8.12...
我從倒數第三行開始就看不懂了
他跟上面的有什麼關係嗎?!
請問有人能解釋看看嗎?
謝謝>"<
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1F：推 bineapple :倒數第三行是要算出你積出來那個sum中每個term的值啊 06/08 17:22
2F：→ bineapple :因為每個term會因為n的奇偶性而變化 所以分開討論 06/08 17:22