Mixing
If $int_0^infty f(x),dx$ exists, then there exists some function $g$ such that $f=o(g)$ and $int_0^infty...
$begingroup$
If $f$ is a positive function such that $int_0^infty f(x),mathrm dx$ converges, show that there is another function $g$ such that $lim_{x to infty} frac {g(x)} {f(x)} = infty$ and $int_0^infty g(x),mathrm dx$ also converges.
integration improper-integrals
edited Jan 11 at 21:19
Did
248k23223460
asked May 25 '13 at 15:41
shmuelshmuel
482
$endgroup$
add a comment |
$begingroup$
If $f$ is a positive function such that $int_0^infty f(x),mathrm dx$ converges, show that there is another function $g$ such that $lim_{x to infty} frac {g(x)} {f(x)} = infty$ and $int_0^infty g(x),mathrm dx$ also converges.
integration improper-integrals
edited Jan 11 at 21:19
Did
248k23223460
asked May 25 '13 at 15:41
shmuelshmuel
482
$endgroup$
add a comment |
$begingroup$
If $f$ is a positive function such that $int_0^infty f(x),mathrm dx$ converges, show that there is another function $g$ such that $lim_{x to infty} frac {g(x)} {f(x)} = infty$ and $int_0^infty g(x),mathrm dx$ also converges.
integration improper-integrals
edited Jan 11 at 21:19
Did
248k23223460
asked May 25 '13 at 15:41
shmuelshmuel
482
$endgroup$
If $f$ is a positive function such that $int_0^infty f(x),mathrm dx$ converges, show that there is another function $g$ such that $lim_{x to infty} frac {g(x)} {f(x)} = infty$ and $int_0^infty g(x),mathrm dx$ also converges.
integration improper-integrals
integration improper-integrals
edited Jan 11 at 21:19
Did
248k23223460
asked May 25 '13 at 15:41
shmuelshmuel
482
edited Jan 11 at 21:19
Did
248k23223460
asked May 25 '13 at 15:41
shmuelshmuel
482
edited Jan 11 at 21:19
Did
248k23223460
edited Jan 11 at 21:19
Did
248k23223460
edited Jan 11 at 21:19
Did
248k23223460
248k23223460
asked May 25 '13 at 15:41
shmuelshmuel
482
asked May 25 '13 at 15:41
shmuelshmuel
482
asked May 25 '13 at 15:41
shmuelshmuel
482
482
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
For every $xgeqslant0$, let $F(x)=int_x^infty f(t),mathrm dt$ and $g(x)=frac{f(x)}{sqrt{F(x)}}$, then:
$int_0^infty g(t),mathrm dt=2sqrt{F(0)}$ is finite,
$f(x)=o(g(x))$ when $xtoinfty$.
These two assertions hold for the same reason, which is that $F(x)to0$ when $xtoinfty$.
Exercise: Adapt this example to show that for every nonnegative sequence $(a_n)$ such that $sumlimits_na_n$ converges there exists some nonnegative sequence $(b_n)$ such that $a_n=o(b_n)$ and $sumlimits_nb_n$ converges.
answered May 25 '13 at 17:03
DidDid
248k23223460
$endgroup$
$begingroup$
Trying to follow: $$int_0^infty g(t) , dt = int_0^infty frac{f(t)}{sqrt{int_t^infty f(s) , ds}} , dt$$
$endgroup$
– Topological cat
Jul 31 '16 at 21:56
$begingroup$
How to get $2sqrt{F(0)}$? Thanks
$endgroup$
– Topological cat
Jul 31 '16 at 21:57
1
$begingroup$
Calculate the derivative of the square root of F
$endgroup$
– Bananach
Jul 31 '16 at 21:59
$begingroup$
@Bananach I see. Thanks
$endgroup$
– Topological cat
Jul 31 '16 at 22:05
add a comment |
Your Answer
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1 Answer
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active
oldest
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1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
For every $xgeqslant0$, let $F(x)=int_x^infty f(t),mathrm dt$ and $g(x)=frac{f(x)}{sqrt{F(x)}}$, then:
$int_0^infty g(t),mathrm dt=2sqrt{F(0)}$ is finite,
$f(x)=o(g(x))$ when $xtoinfty$.
These two assertions hold for the same reason, which is that $F(x)to0$ when $xtoinfty$.
Exercise: Adapt this example to show that for every nonnegative sequence $(a_n)$ such that $sumlimits_na_n$ converges there exists some nonnegative sequence $(b_n)$ such that $a_n=o(b_n)$ and $sumlimits_nb_n$ converges.
answered May 25 '13 at 17:03
DidDid
248k23223460
$endgroup$
$begingroup$
Trying to follow: $$int_0^infty g(t) , dt = int_0^infty frac{f(t)}{sqrt{int_t^infty f(s) , ds}} , dt$$
$endgroup$
– Topological cat
Jul 31 '16 at 21:56
$begingroup$
How to get $2sqrt{F(0)}$? Thanks
$endgroup$
– Topological cat
Jul 31 '16 at 21:57
1
$begingroup$
Calculate the derivative of the square root of F
$endgroup$
– Bananach
Jul 31 '16 at 21:59
$begingroup$
@Bananach I see. Thanks
$endgroup$
– Topological cat
Jul 31 '16 at 22:05
add a comment |
$begingroup$
For every $xgeqslant0$, let $F(x)=int_x^infty f(t),mathrm dt$ and $g(x)=frac{f(x)}{sqrt{F(x)}}$, then:
$int_0^infty g(t),mathrm dt=2sqrt{F(0)}$ is finite,
$f(x)=o(g(x))$ when $xtoinfty$.
These two assertions hold for the same reason, which is that $F(x)to0$ when $xtoinfty$.
Exercise: Adapt this example to show that for every nonnegative sequence $(a_n)$ such that $sumlimits_na_n$ converges there exists some nonnegative sequence $(b_n)$ such that $a_n=o(b_n)$ and $sumlimits_nb_n$ converges.
answered May 25 '13 at 17:03
DidDid
248k23223460
$endgroup$
$begingroup$
Trying to follow: $$int_0^infty g(t) , dt = int_0^infty frac{f(t)}{sqrt{int_t^infty f(s) , ds}} , dt$$
$endgroup$
– Topological cat
Jul 31 '16 at 21:56
$begingroup$
How to get $2sqrt{F(0)}$? Thanks
$endgroup$
– Topological cat
Jul 31 '16 at 21:57
1
$begingroup$
Calculate the derivative of the square root of F
$endgroup$
– Bananach
Jul 31 '16 at 21:59
$begingroup$
@Bananach I see. Thanks
$endgroup$
– Topological cat
Jul 31 '16 at 22:05
add a comment |
$begingroup$
For every $xgeqslant0$, let $F(x)=int_x^infty f(t),mathrm dt$ and $g(x)=frac{f(x)}{sqrt{F(x)}}$, then:
$int_0^infty g(t),mathrm dt=2sqrt{F(0)}$ is finite,
$f(x)=o(g(x))$ when $xtoinfty$.
These two assertions hold for the same reason, which is that $F(x)to0$ when $xtoinfty$.
Exercise: Adapt this example to show that for every nonnegative sequence $(a_n)$ such that $sumlimits_na_n$ converges there exists some nonnegative sequence $(b_n)$ such that $a_n=o(b_n)$ and $sumlimits_nb_n$ converges.
answered May 25 '13 at 17:03
DidDid
248k23223460
$endgroup$
For every $xgeqslant0$, let $F(x)=int_x^infty f(t),mathrm dt$ and $g(x)=frac{f(x)}{sqrt{F(x)}}$, then:
$int_0^infty g(t),mathrm dt=2sqrt{F(0)}$ is finite,
$f(x)=o(g(x))$ when $xtoinfty$.
These two assertions hold for the same reason, which is that $F(x)to0$ when $xtoinfty$.
Exercise: Adapt this example to show that for every nonnegative sequence $(a_n)$ such that $sumlimits_na_n$ converges there exists some nonnegative sequence $(b_n)$ such that $a_n=o(b_n)$ and $sumlimits_nb_n$ converges.
answered May 25 '13 at 17:03
DidDid
248k23223460
edited Jan 11 at 21:19
answered May 25 '13 at 17:03
DidDid
248k23223460
answered May 25 '13 at 17:03
DidDid
248k23223460
answered May 25 '13 at 17:03
DidDid
248k23223460
248k23223460
$begingroup$
Trying to follow: $$int_0^infty g(t) , dt = int_0^infty frac{f(t)}{sqrt{int_t^infty f(s) , ds}} , dt$$
$endgroup$
– Topological cat
Jul 31 '16 at 21:56
$begingroup$
How to get $2sqrt{F(0)}$? Thanks
$endgroup$
– Topological cat
Jul 31 '16 at 21:57
1
$begingroup$
Calculate the derivative of the square root of F
$endgroup$
– Bananach
Jul 31 '16 at 21:59
$begingroup$
@Bananach I see. Thanks
$endgroup$
– Topological cat
Jul 31 '16 at 22:05
add a comment |
$begingroup$
Trying to follow: $$int_0^infty g(t) , dt = int_0^infty frac{f(t)}{sqrt{int_t^infty f(s) , ds}} , dt$$
$endgroup$
– Topological cat
Jul 31 '16 at 21:56
$begingroup$
How to get $2sqrt{F(0)}$? Thanks
$endgroup$
– Topological cat
Jul 31 '16 at 21:57
1
$begingroup$
Calculate the derivative of the square root of F
$endgroup$
– Bananach
Jul 31 '16 at 21:59
$begingroup$
@Bananach I see. Thanks
$endgroup$
– Topological cat
Jul 31 '16 at 22:05
$begingroup$
Trying to follow: $$int_0^infty g(t) , dt = int_0^infty frac{f(t)}{sqrt{int_t^infty f(s) , ds}} , dt$$
$endgroup$
– Topological cat
Jul 31 '16 at 21:56
$begingroup$
Trying to follow: $$int_0^infty g(t) , dt = int_0^infty frac{f(t)}{sqrt{int_t^infty f(s) , ds}} , dt$$
$endgroup$
– Topological cat
Jul 31 '16 at 21:56
$begingroup$
How to get $2sqrt{F(0)}$? Thanks
$endgroup$
– Topological cat
Jul 31 '16 at 21:57
$begingroup$
How to get $2sqrt{F(0)}$? Thanks
$endgroup$
– Topological cat
Jul 31 '16 at 21:57
1
1
$begingroup$
Calculate the derivative of the square root of F
$endgroup$
– Bananach
Jul 31 '16 at 21:59
$begingroup$
Calculate the derivative of the square root of F
$endgroup$
– Bananach
Jul 31 '16 at 21:59
$begingroup$
@Bananach I see. Thanks
$endgroup$
– Topological cat
Jul 31 '16 at 22:05
$begingroup$
@Bananach I see. Thanks
$endgroup$
– Topological cat
Jul 31 '16 at 22:05
add a comment |
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