Under given conditions whether $limlimits_{nto infty}...
$begingroup$
Let ${f_n}_{n=1}^{infty}$ be a sequence of continuous real-valued functions defined on $mathbb R$ which converges pointwise to a continuous real-valued function $f$. Which of the following statements are true?
(i) If $0leq f_n leq f$ for all $nin mathbb N$ then $displaystyle lim_{nto infty} int_{-infty}^{infty}f_n(t)dt=int_{-infty}^{infty}f(t)dt.$
(ii) If $|f_n(t)|leq |sin t|$ for all $tin mathbb R$ and for all $nin mathbb N,$ then $displaystyle lim_{nto infty} int_{-infty}^{infty}f_n(t)dt=int_{-infty}^{infty}f(t)dt.$
(i) If $int_{-infty}^{infty} f<infty$, then we can use DOMINATED CONVERGENCE theorem and can say that the statement is true. But if $int_{-infty}^{infty}f=infty$ then what can we say about the statement?
(ii) I was not able to do this one.
Note: At the answer-key it's given that (i) is true but (ii) is false.
integration sequence-of-function
edited Jan 10 at 16:19
Did
248k23224463
asked Jan 10 at 16:10
nurun neshanurun nesha
1,0362623
$endgroup$
add a comment |
$begingroup$
Let ${f_n}_{n=1}^{infty}$ be a sequence of continuous real-valued functions defined on $mathbb R$ which converges pointwise to a continuous real-valued function $f$. Which of the following statements are true?
(i) If $0leq f_n leq f$ for all $nin mathbb N$ then $displaystyle lim_{nto infty} int_{-infty}^{infty}f_n(t)dt=int_{-infty}^{infty}f(t)dt.$
(ii) If $|f_n(t)|leq |sin t|$ for all $tin mathbb R$ and for all $nin mathbb N,$ then $displaystyle lim_{nto infty} int_{-infty}^{infty}f_n(t)dt=int_{-infty}^{infty}f(t)dt.$
(i) If $int_{-infty}^{infty} f<infty$, then we can use DOMINATED CONVERGENCE theorem and can say that the statement is true. But if $int_{-infty}^{infty}f=infty$ then what can we say about the statement?
(ii) I was not able to do this one.
Note: At the answer-key it's given that (i) is true but (ii) is false.
integration sequence-of-function
edited Jan 10 at 16:19
Did
248k23224463
asked Jan 10 at 16:10
nurun neshanurun nesha
1,0362623
$endgroup$
add a comment |
$begingroup$
Let ${f_n}_{n=1}^{infty}$ be a sequence of continuous real-valued functions defined on $mathbb R$ which converges pointwise to a continuous real-valued function $f$. Which of the following statements are true?
(i) If $0leq f_n leq f$ for all $nin mathbb N$ then $displaystyle lim_{nto infty} int_{-infty}^{infty}f_n(t)dt=int_{-infty}^{infty}f(t)dt.$
(ii) If $|f_n(t)|leq |sin t|$ for all $tin mathbb R$ and for all $nin mathbb N,$ then $displaystyle lim_{nto infty} int_{-infty}^{infty}f_n(t)dt=int_{-infty}^{infty}f(t)dt.$
(i) If $int_{-infty}^{infty} f<infty$, then we can use DOMINATED CONVERGENCE theorem and can say that the statement is true. But if $int_{-infty}^{infty}f=infty$ then what can we say about the statement?
(ii) I was not able to do this one.
Note: At the answer-key it's given that (i) is true but (ii) is false.
integration sequence-of-function
edited Jan 10 at 16:19
Did
248k23224463
asked Jan 10 at 16:10
nurun neshanurun nesha
1,0362623
$endgroup$
Let ${f_n}_{n=1}^{infty}$ be a sequence of continuous real-valued functions defined on $mathbb R$ which converges pointwise to a continuous real-valued function $f$. Which of the following statements are true?
(i) If $0leq f_n leq f$ for all $nin mathbb N$ then $displaystyle lim_{nto infty} int_{-infty}^{infty}f_n(t)dt=int_{-infty}^{infty}f(t)dt.$
(ii) If $|f_n(t)|leq |sin t|$ for all $tin mathbb R$ and for all $nin mathbb N,$ then $displaystyle lim_{nto infty} int_{-infty}^{infty}f_n(t)dt=int_{-infty}^{infty}f(t)dt.$
(i) If $int_{-infty}^{infty} f<infty$, then we can use DOMINATED CONVERGENCE theorem and can say that the statement is true. But if $int_{-infty}^{infty}f=infty$ then what can we say about the statement?
(ii) I was not able to do this one.
Note: At the answer-key it's given that (i) is true but (ii) is false.
integration sequence-of-function
integration sequence-of-function
edited Jan 10 at 16:19
Did
248k23224463
asked Jan 10 at 16:10
nurun neshanurun nesha
1,0362623
edited Jan 10 at 16:19
Did
248k23224463
asked Jan 10 at 16:10
nurun neshanurun nesha
1,0362623
edited Jan 10 at 16:19
Did
248k23224463
edited Jan 10 at 16:19
Did
248k23224463
edited Jan 10 at 16:19
Did
248k23224463
248k23224463
asked Jan 10 at 16:10
nurun neshanurun nesha
1,0362623
asked Jan 10 at 16:10
nurun neshanurun nesha
1,0362623
asked Jan 10 at 16:10
nurun neshanurun nesha
1,0362623
1,0362623
add a comment |
add a comment |
1 Answer
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(1) Suppose $0 le f_n le f$ and $int_{-infty}^infty f(t); dt = infty$. Given $N > 0$, there is $M$ such that $int_{-M}^M f(t); dt > N$. By dominated convergence $int_{-M}^M f_n(t); dt to int_{-M}^M f(t); dt$, so $int_{-infty}^infty f_n(t); dt ge int_{-M}^M f_n(t); dt > N$ for sufficiently large $n$.
(2) Try $f_n(t) = sin(t)$ for $n pi < t < (n+1)pi$, $0$ otherwise.
answered Jan 10 at 16:20
Robert IsraelRobert Israel
326k23215469
$endgroup$
3
$begingroup$
$f$ is stated to be a continuous real-valued function.
$endgroup$
– Robert Israel
Jan 10 at 16:25
add a comment |
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1 Answer
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1 Answer
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$begingroup$
(1) Suppose $0 le f_n le f$ and $int_{-infty}^infty f(t); dt = infty$. Given $N > 0$, there is $M$ such that $int_{-M}^M f(t); dt > N$. By dominated convergence $int_{-M}^M f_n(t); dt to int_{-M}^M f(t); dt$, so $int_{-infty}^infty f_n(t); dt ge int_{-M}^M f_n(t); dt > N$ for sufficiently large $n$.
(2) Try $f_n(t) = sin(t)$ for $n pi < t < (n+1)pi$, $0$ otherwise.
answered Jan 10 at 16:20
Robert IsraelRobert Israel
326k23215469
$endgroup$
3
$begingroup$
$f$ is stated to be a continuous real-valued function.
$endgroup$
– Robert Israel
Jan 10 at 16:25
add a comment |
$begingroup$
(1) Suppose $0 le f_n le f$ and $int_{-infty}^infty f(t); dt = infty$. Given $N > 0$, there is $M$ such that $int_{-M}^M f(t); dt > N$. By dominated convergence $int_{-M}^M f_n(t); dt to int_{-M}^M f(t); dt$, so $int_{-infty}^infty f_n(t); dt ge int_{-M}^M f_n(t); dt > N$ for sufficiently large $n$.
(2) Try $f_n(t) = sin(t)$ for $n pi < t < (n+1)pi$, $0$ otherwise.
answered Jan 10 at 16:20
Robert IsraelRobert Israel
326k23215469
$endgroup$
3
$begingroup$
$f$ is stated to be a continuous real-valued function.
$endgroup$
– Robert Israel
Jan 10 at 16:25
add a comment |
$begingroup$
(1) Suppose $0 le f_n le f$ and $int_{-infty}^infty f(t); dt = infty$. Given $N > 0$, there is $M$ such that $int_{-M}^M f(t); dt > N$. By dominated convergence $int_{-M}^M f_n(t); dt to int_{-M}^M f(t); dt$, so $int_{-infty}^infty f_n(t); dt ge int_{-M}^M f_n(t); dt > N$ for sufficiently large $n$.
(2) Try $f_n(t) = sin(t)$ for $n pi < t < (n+1)pi$, $0$ otherwise.
answered Jan 10 at 16:20
Robert IsraelRobert Israel
326k23215469
$endgroup$
(1) Suppose $0 le f_n le f$ and $int_{-infty}^infty f(t); dt = infty$. Given $N > 0$, there is $M$ such that $int_{-M}^M f(t); dt > N$. By dominated convergence $int_{-M}^M f_n(t); dt to int_{-M}^M f(t); dt$, so $int_{-infty}^infty f_n(t); dt ge int_{-M}^M f_n(t); dt > N$ for sufficiently large $n$.
(2) Try $f_n(t) = sin(t)$ for $n pi < t < (n+1)pi$, $0$ otherwise.
answered Jan 10 at 16:20
Robert IsraelRobert Israel
326k23215469
answered Jan 10 at 16:20
Robert IsraelRobert Israel
326k23215469
answered Jan 10 at 16:20
Robert IsraelRobert Israel
326k23215469
answered Jan 10 at 16:20
Robert IsraelRobert Israel
326k23215469
326k23215469
3
$begingroup$
$f$ is stated to be a continuous real-valued function.
$endgroup$
– Robert Israel
Jan 10 at 16:25
add a comment |
3
$begingroup$
$f$ is stated to be a continuous real-valued function.
$endgroup$
– Robert Israel
Jan 10 at 16:25
3
3
$begingroup$
$f$ is stated to be a continuous real-valued function.
$endgroup$
– Robert Israel
Jan 10 at 16:25
$begingroup$
$f$ is stated to be a continuous real-valued function.
$endgroup$
– Robert Israel
Jan 10 at 16:25
add a comment |
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