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Homework Statement
f(x) is a continuous and positive function when [tex] x\in[0,\infty)[/tex]. (#1)
[tex] x_n [/tex] is a monotonic increasing sequence, [tex]x_0=0[/tex] [tex],x_n \rightarrow \infty[/tex]. (#2)
Prove or contradict:
[tex] \mbox{If } \sum_{n=0}^\infty \int_{x_n}^{x_(n+1)} f(x)dx \mbox{ is convergent (#3) then } \int_{0}^\infty f(x)dx \mbox{ is also convergent.}[/tex]3. The attempt
[tex](*3)\ and\ by\ the\ Cauchy\ Criterion\ \Longrightarrow\ \forall\ \epsilon>0\ \exists\ N_1>0,\ so\ \forall\ m>k>N_1[/tex]
[tex]\left \int_{x_(k+1)}^{x_(m+1)} f(x)dx=(*1)=| \sum_{n=0}^\infty \int_{x_n}^{x_(n+1)} f(x)dx|<\epsilon\mbox{ (*4)} \right [/tex][tex](*2)\ \Longrightarrow\ \forall\ N_1>0\ \exists\ N>0\ so\ \forall\ n>N_1,\ x_n>N\ \ \ \ (*5) } [/tex]
[tex](*4)\ and\ (*5)\ \Longrightarrow\ \forall\ m>k>N\ \int_{x_(k+1)}^{x_(m+1)} f(x)dx<\epsilon\\\Longrightarrow\ Cauchy\ Criterion\ \int_{0}^\infty f(x)dx\ is\ convergent. [/tex]
It seems right to me, but I'm not sure...
I think i also have vice versa proof.
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