Re: [微積] 請教兩題Improper Integral的證明
※ 引述《bestscott (新希望)》之銘言：
: f(x)=1 在區間[1,1+1/2],[2,2+1/4],[3,3+1/8],...,and f(x)=0 elsewhere
: (1)show that
: ∞
: ∫ f(x)dx converges(and is equal to 1) although f(x)不趨近於0,x→∞.
: 0
∞ ∞ n+(1/2^n) ∞
Since ∫ f(x)dx = Σ ∫ 1 dx = Σ 1/2^n = 1
0 n=1 n n=1
But lim f(x) does not exist.
x->∞
(Take x = n and x = n + 1/(n-1) to get the differente limit.)
: (2) ∞
: Modify f to make an example of a function g such that∫0 g(x)dx converges
: although g(x) does not remain bounded as x→∞.
: 謝謝!!! QQ
Consider
┌ n, if x in [n,n+(1/2^n)], n = 1,2,3,...
g(x) = ┴ 0, elsewhere
Obviously, g(x) is unbounded as x->∞
∞ ∞ n+(1/2^n) ∞
∫ g(x)dx = Σ ∫ n dx = Σ n/2^n
0 n=1 n n=1
Cleraly, this series converges by root test, hence, the integral converge.
And the value of the series is easy to calculate, try yourself!
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