A convergent improper integral is non-zero?
$begingroup$
Consider a function $f(x):mathbb{R}mapsto mathbb{R}$. If the improper integral
begin{align}
int_{0}^{infty}f(x)dx
end{align}
converges, will the value of this integral be non-zero?
Or are there examples of $f(x)$ with improper integral being zero. (except for $f(x)=0$ for all $x$)
real-analysis integration
asked Dec 20 '18 at 3:41
guluzhuguluzhu
538
$endgroup$
add a comment |
$begingroup$
Consider a function $f(x):mathbb{R}mapsto mathbb{R}$. If the improper integral
begin{align}
int_{0}^{infty}f(x)dx
end{align}
converges, will the value of this integral be non-zero?
Or are there examples of $f(x)$ with improper integral being zero. (except for $f(x)=0$ for all $x$)
real-analysis integration
asked Dec 20 '18 at 3:41
guluzhuguluzhu
538
$endgroup$
$begingroup$
Let $sum_{n=0}^{infty} a_n$ be your favorite series that converges to zero. Define $$f(x) = sum_{n=0}^{infty}a_n g(x - n),$$ where $$g(x) = begin{cases} 1 & text{if $0 leq x < 1$} \ 0 & text{otherwise}end{cases}$$
$endgroup$
– Bungo
Dec 20 '18 at 6:01
add a comment |
$begingroup$
Consider a function $f(x):mathbb{R}mapsto mathbb{R}$. If the improper integral
begin{align}
int_{0}^{infty}f(x)dx
end{align}
converges, will the value of this integral be non-zero?
Or are there examples of $f(x)$ with improper integral being zero. (except for $f(x)=0$ for all $x$)
real-analysis integration
asked Dec 20 '18 at 3:41
guluzhuguluzhu
538
$endgroup$
Consider a function $f(x):mathbb{R}mapsto mathbb{R}$. If the improper integral
begin{align}
int_{0}^{infty}f(x)dx
end{align}
converges, will the value of this integral be non-zero?
Or are there examples of $f(x)$ with improper integral being zero. (except for $f(x)=0$ for all $x$)
real-analysis integration
real-analysis integration
asked Dec 20 '18 at 3:41
guluzhuguluzhu
538
asked Dec 20 '18 at 3:41
guluzhuguluzhu
538
edited Dec 20 '18 at 3:54
guluzhu
asked Dec 20 '18 at 3:41
guluzhuguluzhu
538
asked Dec 20 '18 at 3:41
guluzhuguluzhu
538
asked Dec 20 '18 at 3:41
guluzhuguluzhu
538
538
$begingroup$
Let $sum_{n=0}^{infty} a_n$ be your favorite series that converges to zero. Define $$f(x) = sum_{n=0}^{infty}a_n g(x - n),$$ where $$g(x) = begin{cases} 1 & text{if $0 leq x < 1$} \ 0 & text{otherwise}end{cases}$$
$endgroup$
– Bungo
Dec 20 '18 at 6:01
add a comment |
$begingroup$
Let $sum_{n=0}^{infty} a_n$ be your favorite series that converges to zero. Define $$f(x) = sum_{n=0}^{infty}a_n g(x - n),$$ where $$g(x) = begin{cases} 1 & text{if $0 leq x < 1$} \ 0 & text{otherwise}end{cases}$$
$endgroup$
– Bungo
Dec 20 '18 at 6:01
$begingroup$
Let $sum_{n=0}^{infty} a_n$ be your favorite series that converges to zero. Define $$f(x) = sum_{n=0}^{infty}a_n g(x - n),$$ where $$g(x) = begin{cases} 1 & text{if $0 leq x < 1$} \ 0 & text{otherwise}end{cases}$$
$endgroup$
– Bungo
Dec 20 '18 at 6:01
$begingroup$
Let $sum_{n=0}^{infty} a_n$ be your favorite series that converges to zero. Define $$f(x) = sum_{n=0}^{infty}a_n g(x - n),$$ where $$g(x) = begin{cases} 1 & text{if $0 leq x < 1$} \ 0 & text{otherwise}end{cases}$$
$endgroup$
– Bungo
Dec 20 '18 at 6:01
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
$$int_0^{infty } left(frac{1}{(x+1)^2}-frac{2}{(x+1)^3}right)dx=0$$
answered Dec 20 '18 at 3:45
Kemono ChenKemono Chen
3,1991844
$endgroup$
add a comment |
$begingroup$
Take any $g$ for which the improper integral $int_0^{infty} g(x)dx$ has some finite value $C$ and take $f(x)=g(x)-Ce^{-x}$. Examples of $g$ are plenty; for example $g(x)=e^{-ax}$ with $a>0$.
answered Dec 20 '18 at 5:40
Kavi Rama MurthyKavi Rama Murthy
69.1k53169
$endgroup$
add a comment |
$begingroup$
Here is another example of an improper integral being zero:
begin{align}
int_0^infty frac{log x}{x^2+1},dx=0
end{align}
answered Dec 20 '18 at 5:55
ZacharyZachary
2,3751214
$endgroup$
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
$$int_0^{infty } left(frac{1}{(x+1)^2}-frac{2}{(x+1)^3}right)dx=0$$
answered Dec 20 '18 at 3:45
Kemono ChenKemono Chen
3,1991844
$endgroup$
add a comment |
$begingroup$
$$int_0^{infty } left(frac{1}{(x+1)^2}-frac{2}{(x+1)^3}right)dx=0$$
answered Dec 20 '18 at 3:45
Kemono ChenKemono Chen
3,1991844
$endgroup$
add a comment |
$begingroup$
$$int_0^{infty } left(frac{1}{(x+1)^2}-frac{2}{(x+1)^3}right)dx=0$$
answered Dec 20 '18 at 3:45
Kemono ChenKemono Chen
3,1991844
$endgroup$
$$int_0^{infty } left(frac{1}{(x+1)^2}-frac{2}{(x+1)^3}right)dx=0$$
answered Dec 20 '18 at 3:45
Kemono ChenKemono Chen
3,1991844
answered Dec 20 '18 at 3:45
Kemono ChenKemono Chen
3,1991844
answered Dec 20 '18 at 3:45
Kemono ChenKemono Chen
3,1991844
answered Dec 20 '18 at 3:45
Kemono ChenKemono Chen
3,1991844
3,1991844
add a comment |
add a comment |
$begingroup$
Take any $g$ for which the improper integral $int_0^{infty} g(x)dx$ has some finite value $C$ and take $f(x)=g(x)-Ce^{-x}$. Examples of $g$ are plenty; for example $g(x)=e^{-ax}$ with $a>0$.
answered Dec 20 '18 at 5:40
Kavi Rama MurthyKavi Rama Murthy
69.1k53169
$endgroup$
add a comment |
$begingroup$
Take any $g$ for which the improper integral $int_0^{infty} g(x)dx$ has some finite value $C$ and take $f(x)=g(x)-Ce^{-x}$. Examples of $g$ are plenty; for example $g(x)=e^{-ax}$ with $a>0$.
answered Dec 20 '18 at 5:40
Kavi Rama MurthyKavi Rama Murthy
69.1k53169
$endgroup$
add a comment |
$begingroup$
Take any $g$ for which the improper integral $int_0^{infty} g(x)dx$ has some finite value $C$ and take $f(x)=g(x)-Ce^{-x}$. Examples of $g$ are plenty; for example $g(x)=e^{-ax}$ with $a>0$.
answered Dec 20 '18 at 5:40
Kavi Rama MurthyKavi Rama Murthy
69.1k53169
$endgroup$
Take any $g$ for which the improper integral $int_0^{infty} g(x)dx$ has some finite value $C$ and take $f(x)=g(x)-Ce^{-x}$. Examples of $g$ are plenty; for example $g(x)=e^{-ax}$ with $a>0$.
answered Dec 20 '18 at 5:40
Kavi Rama MurthyKavi Rama Murthy
69.1k53169
answered Dec 20 '18 at 5:40
Kavi Rama MurthyKavi Rama Murthy
69.1k53169
answered Dec 20 '18 at 5:40
Kavi Rama MurthyKavi Rama Murthy
69.1k53169
answered Dec 20 '18 at 5:40
Kavi Rama MurthyKavi Rama Murthy
69.1k53169
69.1k53169
add a comment |
add a comment |
$begingroup$
Here is another example of an improper integral being zero:
begin{align}
int_0^infty frac{log x}{x^2+1},dx=0
end{align}
answered Dec 20 '18 at 5:55
ZacharyZachary
2,3751214
$endgroup$
add a comment |
$begingroup$
Here is another example of an improper integral being zero:
begin{align}
int_0^infty frac{log x}{x^2+1},dx=0
end{align}
answered Dec 20 '18 at 5:55
ZacharyZachary
2,3751214
$endgroup$
add a comment |
$begingroup$
Here is another example of an improper integral being zero:
begin{align}
int_0^infty frac{log x}{x^2+1},dx=0
end{align}
answered Dec 20 '18 at 5:55
ZacharyZachary
2,3751214
$endgroup$
Here is another example of an improper integral being zero:
begin{align}
int_0^infty frac{log x}{x^2+1},dx=0
end{align}
answered Dec 20 '18 at 5:55
ZacharyZachary
2,3751214
answered Dec 20 '18 at 5:55
ZacharyZachary
2,3751214
answered Dec 20 '18 at 5:55
ZacharyZachary
2,3751214
answered Dec 20 '18 at 5:55
ZacharyZachary
2,3751214
2,3751214
add a comment |
add a comment |
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$begingroup$
Let $sum_{n=0}^{infty} a_n$ be your favorite series that converges to zero. Define $$f(x) = sum_{n=0}^{infty}a_n g(x - n),$$ where $$g(x) = begin{cases} 1 & text{if $0 leq x < 1$} \ 0 & text{otherwise}end{cases}$$
$endgroup$
– Bungo
Dec 20 '18 at 6:01