作者wind0630 (小松鼠)
看板Math
标题[微积] 一题高微有地方不懂
时间Wed Jun 8 15:55:49 2011
Let f(x)=Σn^(-2)sinnx.Show that f is a continuous function on R
(1~无限大)
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~无限大) (0~拍/2) (1~无限大)
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...
我从倒数第三行开始就看不懂了
他跟上面的有什麽关系吗?!
请问有人能解释看看吗?
谢谢>"<
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1F：推 bineapple :倒数第三行是要算出你积出来那个sum中每个term的值啊 06/08 17:22
2F：→ bineapple :因为每个term会因为n的奇偶性而变化 所以分开讨论 06/08 17:22