Mixing
Question over limit of integrand for definite integral.
$begingroup$
Based on this question, a question (for myself) arose about how limits of a definite integral are evaluated.
The limit integral takes the form:
begin{equation}
L = lim_{n rightarrow N} int_{a}^{b} f_n(x):dx
end{equation}
Now if $f_n(x)$ has a primitive $F_n(x)$ then:
begin{equation}
L = lim_{n rightarrow N} int_{a}^{b} f_n(x):dx = lim_{n rightarrow N} big[ F_n(a) - F_n(b) big]
end{equation}
My question is: Is the resultant limit expression equivalent to the integral with the limit applied to the integrand, i.e. is
begin{equation}
L = lim_{n rightarrow N} int_{a}^{b} f_n(x):dx = int_{a}^{b} lim_{n rightarrow N} :f_n(x):dx
end{equation}
I've done some googling around and I haven't been able to find anything. Here impose that $f_n(x)$ is continuous over $[a,b]$.
real-analysis integration limits definite-integrals
asked Jan 9 at 2:17
DavidGDavidG
2,2651721
$endgroup$
add a comment |
$begingroup$
Based on this question, a question (for myself) arose about how limits of a definite integral are evaluated.
The limit integral takes the form:
begin{equation}
L = lim_{n rightarrow N} int_{a}^{b} f_n(x):dx
end{equation}
Now if $f_n(x)$ has a primitive $F_n(x)$ then:
begin{equation}
L = lim_{n rightarrow N} int_{a}^{b} f_n(x):dx = lim_{n rightarrow N} big[ F_n(a) - F_n(b) big]
end{equation}
My question is: Is the resultant limit expression equivalent to the integral with the limit applied to the integrand, i.e. is
begin{equation}
L = lim_{n rightarrow N} int_{a}^{b} f_n(x):dx = int_{a}^{b} lim_{n rightarrow N} :f_n(x):dx
end{equation}
I've done some googling around and I haven't been able to find anything. Here impose that $f_n(x)$ is continuous over $[a,b]$.
real-analysis integration limits definite-integrals
asked Jan 9 at 2:17
DavidGDavidG
2,2651721
$endgroup$
3
$begingroup$
Look up the Dominated Convergence Theorem.
$endgroup$
– D.B.
Jan 9 at 2:19
add a comment |
$begingroup$
Based on this question, a question (for myself) arose about how limits of a definite integral are evaluated.
The limit integral takes the form:
begin{equation}
L = lim_{n rightarrow N} int_{a}^{b} f_n(x):dx
end{equation}
Now if $f_n(x)$ has a primitive $F_n(x)$ then:
begin{equation}
L = lim_{n rightarrow N} int_{a}^{b} f_n(x):dx = lim_{n rightarrow N} big[ F_n(a) - F_n(b) big]
end{equation}
My question is: Is the resultant limit expression equivalent to the integral with the limit applied to the integrand, i.e. is
begin{equation}
L = lim_{n rightarrow N} int_{a}^{b} f_n(x):dx = int_{a}^{b} lim_{n rightarrow N} :f_n(x):dx
end{equation}
I've done some googling around and I haven't been able to find anything. Here impose that $f_n(x)$ is continuous over $[a,b]$.
real-analysis integration limits definite-integrals
asked Jan 9 at 2:17
DavidGDavidG
2,2651721
$endgroup$
Based on this question, a question (for myself) arose about how limits of a definite integral are evaluated.
The limit integral takes the form:
begin{equation}
L = lim_{n rightarrow N} int_{a}^{b} f_n(x):dx
end{equation}
Now if $f_n(x)$ has a primitive $F_n(x)$ then:
begin{equation}
L = lim_{n rightarrow N} int_{a}^{b} f_n(x):dx = lim_{n rightarrow N} big[ F_n(a) - F_n(b) big]
end{equation}
My question is: Is the resultant limit expression equivalent to the integral with the limit applied to the integrand, i.e. is
begin{equation}
L = lim_{n rightarrow N} int_{a}^{b} f_n(x):dx = int_{a}^{b} lim_{n rightarrow N} :f_n(x):dx
end{equation}
I've done some googling around and I haven't been able to find anything. Here impose that $f_n(x)$ is continuous over $[a,b]$.
real-analysis integration limits definite-integrals
real-analysis integration limits definite-integrals
asked Jan 9 at 2:17
DavidGDavidG
2,2651721
asked Jan 9 at 2:17
DavidGDavidG
2,2651721
asked Jan 9 at 2:17
DavidGDavidG
2,2651721
asked Jan 9 at 2:17
DavidGDavidG
2,2651721
asked Jan 9 at 2:17
DavidGDavidG
2,2651721
2,2651721
3
$begingroup$
Look up the Dominated Convergence Theorem.
$endgroup$
– D.B.
Jan 9 at 2:19
add a comment |
3
$begingroup$
Look up the Dominated Convergence Theorem.
$endgroup$
– D.B.
Jan 9 at 2:19
3
3
$begingroup$
Look up the Dominated Convergence Theorem.
$endgroup$
– D.B.
Jan 9 at 2:19
$begingroup$
Look up the Dominated Convergence Theorem.
$endgroup$
– D.B.
Jan 9 at 2:19
add a comment |
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$begingroup$
Look up the Dominated Convergence Theorem.
$endgroup$
– D.B.
Jan 9 at 2:19