Find the limit of $int_{0}^{pi }frac{sin nx}{nx} mathrm dx $.
If $a>0,$ show that $lim limits_{n to infty} int_{ a}^{pi} frac{sin nx}{nx} mathrm dx =0,$ what happens if $a=0.$
The first part is easy since the integrand uniformly converges to zero on $[a, pi]$ and for each $n, ; f_n(x)= frac {sin nx}{nx}$ is continuous on $[a,pi].$
But for $a=0$ I don't understand how to proceed. I was thinking of using bounded convergence theorem but that required uniform boundedness of the integrand .
Please give me a hint. Thanks in advance.
real-analysis
asked Nov 23 at 19:28
Normal
647
add a comment |
If $a>0,$ show that $lim limits_{n to infty} int_{ a}^{pi} frac{sin nx}{nx} mathrm dx =0,$ what happens if $a=0.$
The first part is easy since the integrand uniformly converges to zero on $[a, pi]$ and for each $n, ; f_n(x)= frac {sin nx}{nx}$ is continuous on $[a,pi].$
But for $a=0$ I don't understand how to proceed. I was thinking of using bounded convergence theorem but that required uniform boundedness of the integrand .
Please give me a hint. Thanks in advance.
real-analysis
asked Nov 23 at 19:28
Normal
647
is this what you aimed to write? Please check.
– user376343
Nov 23 at 19:34
1
Where is $a$ in the integral?
– Will M.
Nov 23 at 19:34
No, sorry, I fixed it
– Normal
Nov 23 at 19:38
The integrand is continuous and bounded, so what happens when you integrate on $[0,a]$ with $ato0$?
– Jean-Claude Arbaut
Nov 23 at 19:45
The integrand is continuous on [0,pi] considering $ f_n(0) $=1 and hence it is bounded on [0,pi]. Sorry I am a beginner in real analysis ..I could not understand what you said .
– Normal
Nov 23 at 20:17
add a comment |
If $a>0,$ show that $lim limits_{n to infty} int_{ a}^{pi} frac{sin nx}{nx} mathrm dx =0,$ what happens if $a=0.$
The first part is easy since the integrand uniformly converges to zero on $[a, pi]$ and for each $n, ; f_n(x)= frac {sin nx}{nx}$ is continuous on $[a,pi].$
But for $a=0$ I don't understand how to proceed. I was thinking of using bounded convergence theorem but that required uniform boundedness of the integrand .
Please give me a hint. Thanks in advance.
real-analysis
asked Nov 23 at 19:28
Normal
647
If $a>0,$ show that $lim limits_{n to infty} int_{ a}^{pi} frac{sin nx}{nx} mathrm dx =0,$ what happens if $a=0.$
The first part is easy since the integrand uniformly converges to zero on $[a, pi]$ and for each $n, ; f_n(x)= frac {sin nx}{nx}$ is continuous on $[a,pi].$
But for $a=0$ I don't understand how to proceed. I was thinking of using bounded convergence theorem but that required uniform boundedness of the integrand .
Please give me a hint. Thanks in advance.
real-analysis
real-analysis
asked Nov 23 at 19:28
Normal
647
asked Nov 23 at 19:28
Normal
647
edited Nov 23 at 19:35
asked Nov 23 at 19:28
Normal
647
asked Nov 23 at 19:28
Normal
647
asked Nov 23 at 19:28
Normal
647
647
is this what you aimed to write? Please check.
– user376343
Nov 23 at 19:34
1
Where is $a$ in the integral?
– Will M.
Nov 23 at 19:34
No, sorry, I fixed it
– Normal
Nov 23 at 19:38
The integrand is continuous and bounded, so what happens when you integrate on $[0,a]$ with $ato0$?
– Jean-Claude Arbaut
Nov 23 at 19:45
The integrand is continuous on [0,pi] considering $ f_n(0) $=1 and hence it is bounded on [0,pi]. Sorry I am a beginner in real analysis ..I could not understand what you said .
– Normal
Nov 23 at 20:17
add a comment |
is this what you aimed to write? Please check.
– user376343
Nov 23 at 19:34
1
Where is $a$ in the integral?
– Will M.
Nov 23 at 19:34
No, sorry, I fixed it
– Normal
Nov 23 at 19:38
The integrand is continuous and bounded, so what happens when you integrate on $[0,a]$ with $ato0$?
– Jean-Claude Arbaut
Nov 23 at 19:45
The integrand is continuous on [0,pi] considering $ f_n(0) $=1 and hence it is bounded on [0,pi]. Sorry I am a beginner in real analysis ..I could not understand what you said .
– Normal
Nov 23 at 20:17
is this what you aimed to write? Please check.
– user376343
Nov 23 at 19:34
is this what you aimed to write? Please check.
– user376343
Nov 23 at 19:34
1
1
Where is $a$ in the integral?
– Will M.
Nov 23 at 19:34
Where is $a$ in the integral?
– Will M.
Nov 23 at 19:34
No, sorry, I fixed it
– Normal
Nov 23 at 19:38
No, sorry, I fixed it
– Normal
Nov 23 at 19:38
The integrand is continuous and bounded, so what happens when you integrate on $[0,a]$ with $ato0$?
– Jean-Claude Arbaut
Nov 23 at 19:45
The integrand is continuous and bounded, so what happens when you integrate on $[0,a]$ with $ato0$?
– Jean-Claude Arbaut
Nov 23 at 19:45
The integrand is continuous on [0,pi] considering $ f_n(0) $=1 and hence it is bounded on [0,pi]. Sorry I am a beginner in real analysis ..I could not understand what you said .
– Normal
Nov 23 at 20:17
The integrand is continuous on [0,pi] considering $ f_n(0) $=1 and hence it is bounded on [0,pi]. Sorry I am a beginner in real analysis ..I could not understand what you said .
– Normal
Nov 23 at 20:17
add a comment |
2 Answers
2
active
oldest
votes
For all $x$, $|f_n(x)|le1$, hence
$$left|int_0^{a}f_n(x)dxright|le (a-0) sup_{[0,a]}|f_n|le a$$
Therefore
$$left|int_0^pi f_n(x)dxright|=left|int_0^a f_n(x)dx+int_a^pi f_n(x)dxright|le a+left|int_a^pi f_n(x)dxright|$$
Now, for all $epsilon>0$, let $a=epsilon$, and the remaining integral term tends to $0$ as $nto0$ (as $a$ is fixed as soon as $epsilon$ is fixed), so there is a $N$ such that for all $n>N$, $left|int_a^pi f_n(x)dxright|<epsilon$.
Hence, for all $epsilon>0$, there is a $N$ such that for all $n>N$, $left|int_0^pi f_n(x)dxright|<2epsilon$.
Hence $int_0^pi f_n(x)dxto0$ as $ntoinfty$.
answered Nov 23 at 21:15
Jean-Claude Arbaut
14.7k63464
Thank you ..I have a question....first the given sequences of function are not defined at 0 but since the limit exists at zero and the limiting value is 1 , I can define $ f_n(0) $=1, so did you write $ | f_n(x)| le 1$ with the same thought I have or something else ..If there is an other explanation please tell me ....so then $ f_n(x) $ is uniformly bounded and I can use bounded convergence theorem to conclude that the limiting value is zero..can I do that ?
– Normal
Nov 23 at 22:19
@Normal Yes, you can define $f_n(0)=1$ because the limit exists. It's a removable singularity, and the function thus defined is $C^{infty}$. See also en.wikipedia.org/wiki/Sinc_function
– Jean-Claude Arbaut
Nov 23 at 22:21
Thank you... please tell me it will be ok if I use bounded convergence theorem by using the uniform boundedness of $ f_n(x) $
– Normal
Nov 23 at 22:38
@Normal Yes, it also works. But you're not really a beginner then :)
– Jean-Claude Arbaut
Nov 23 at 22:41
One more question what if I choose an $ epsilon $ different from a .
– Normal
Nov 23 at 23:05
|
show 2 more comments
We have that $$int_{0}^{pi }frac{sin nx}{nx} mathrm dx=int_{0}^{npi }frac{sin nx}{nx} mathrm {1over n}dnx={1over n }int_{0}^{npi} {sin xover x}dx$$also we know that $$int_0^{infty}{sin xover x}dx={pi over 2}$$according to Integration of Sinc function therefore $$lim _{nto infty}int_{0}^{pi }frac{sin nx}{nx} mathrm dx=0$$
answered Nov 23 at 19:43
Mostafa Ayaz
13.7k3836
Thank you .So $ lim_{nto infty} int_{0}^{pi} frac{sin nx}{nx} $=$( lim_{nto infty} {1 over n})(lim_{nto infty} int_{0}^{ npi}{sin x over x}) dx $ =0×(pi/2)=0. Is it what you did?
– Normal
Nov 23 at 20:05
You're welcome. Also could be explained that way. Yes...
– Mostafa Ayaz
Nov 23 at 20:10
Sorry for disturbing you again ,I have a doubt am I allowed to write what I wrote above..I mean I know lim ($ x_n y_n $ ) = lim ($ x_n $)lim ( $ y_n $) when the both limit exists ..but am I allowed to do the same thing in here.
– Normal
Nov 23 at 21:18
No mind at all :) Notice that i concluded that $$0le lim int_{0}^{pi }frac{sin nx}{nx} mathrm dx= lim {1over n }int_{0}^{npi} {sin xover x}dxlelim {1over n }int_{0}^{infty} {sin xover x}dx=lim {pi over 2n}=0$$ and used squeeze theorem
– Mostafa Ayaz
Nov 23 at 21:21
Thanks for answering though I didn't understand the line where you wrote lim $ {1 over n} int_{0}^{npi} { sin x over x} dx le lim {1 over n} int_{0}^{infty} {sin x over x} dx $ but it's alright the function is uniformly bounded defining $ f_n(0) =1 $ so by bounded convergence theorem I can conclude the answer is zero .
– Normal
Nov 23 at 22:31
|
show 1 more comment
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2 Answers
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2 Answers
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oldest
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For all $x$, $|f_n(x)|le1$, hence
$$left|int_0^{a}f_n(x)dxright|le (a-0) sup_{[0,a]}|f_n|le a$$
Therefore
$$left|int_0^pi f_n(x)dxright|=left|int_0^a f_n(x)dx+int_a^pi f_n(x)dxright|le a+left|int_a^pi f_n(x)dxright|$$
Now, for all $epsilon>0$, let $a=epsilon$, and the remaining integral term tends to $0$ as $nto0$ (as $a$ is fixed as soon as $epsilon$ is fixed), so there is a $N$ such that for all $n>N$, $left|int_a^pi f_n(x)dxright|<epsilon$.
Hence, for all $epsilon>0$, there is a $N$ such that for all $n>N$, $left|int_0^pi f_n(x)dxright|<2epsilon$.
Hence $int_0^pi f_n(x)dxto0$ as $ntoinfty$.
answered Nov 23 at 21:15
Jean-Claude Arbaut
14.7k63464
Thank you ..I have a question....first the given sequences of function are not defined at 0 but since the limit exists at zero and the limiting value is 1 , I can define $ f_n(0) $=1, so did you write $ | f_n(x)| le 1$ with the same thought I have or something else ..If there is an other explanation please tell me ....so then $ f_n(x) $ is uniformly bounded and I can use bounded convergence theorem to conclude that the limiting value is zero..can I do that ?
– Normal
Nov 23 at 22:19
@Normal Yes, you can define $f_n(0)=1$ because the limit exists. It's a removable singularity, and the function thus defined is $C^{infty}$. See also en.wikipedia.org/wiki/Sinc_function
– Jean-Claude Arbaut
Nov 23 at 22:21
Thank you... please tell me it will be ok if I use bounded convergence theorem by using the uniform boundedness of $ f_n(x) $
– Normal
Nov 23 at 22:38
@Normal Yes, it also works. But you're not really a beginner then :)
– Jean-Claude Arbaut
Nov 23 at 22:41
One more question what if I choose an $ epsilon $ different from a .
– Normal
Nov 23 at 23:05
|
show 2 more comments
For all $x$, $|f_n(x)|le1$, hence
$$left|int_0^{a}f_n(x)dxright|le (a-0) sup_{[0,a]}|f_n|le a$$
Therefore
$$left|int_0^pi f_n(x)dxright|=left|int_0^a f_n(x)dx+int_a^pi f_n(x)dxright|le a+left|int_a^pi f_n(x)dxright|$$
Now, for all $epsilon>0$, let $a=epsilon$, and the remaining integral term tends to $0$ as $nto0$ (as $a$ is fixed as soon as $epsilon$ is fixed), so there is a $N$ such that for all $n>N$, $left|int_a^pi f_n(x)dxright|<epsilon$.
Hence, for all $epsilon>0$, there is a $N$ such that for all $n>N$, $left|int_0^pi f_n(x)dxright|<2epsilon$.
Hence $int_0^pi f_n(x)dxto0$ as $ntoinfty$.
answered Nov 23 at 21:15
Jean-Claude Arbaut
14.7k63464
Thank you ..I have a question....first the given sequences of function are not defined at 0 but since the limit exists at zero and the limiting value is 1 , I can define $ f_n(0) $=1, so did you write $ | f_n(x)| le 1$ with the same thought I have or something else ..If there is an other explanation please tell me ....so then $ f_n(x) $ is uniformly bounded and I can use bounded convergence theorem to conclude that the limiting value is zero..can I do that ?
– Normal
Nov 23 at 22:19
@Normal Yes, you can define $f_n(0)=1$ because the limit exists. It's a removable singularity, and the function thus defined is $C^{infty}$. See also en.wikipedia.org/wiki/Sinc_function
– Jean-Claude Arbaut
Nov 23 at 22:21
Thank you... please tell me it will be ok if I use bounded convergence theorem by using the uniform boundedness of $ f_n(x) $
– Normal
Nov 23 at 22:38
@Normal Yes, it also works. But you're not really a beginner then :)
– Jean-Claude Arbaut
Nov 23 at 22:41
One more question what if I choose an $ epsilon $ different from a .
– Normal
Nov 23 at 23:05
|
show 2 more comments
For all $x$, $|f_n(x)|le1$, hence
$$left|int_0^{a}f_n(x)dxright|le (a-0) sup_{[0,a]}|f_n|le a$$
Therefore
$$left|int_0^pi f_n(x)dxright|=left|int_0^a f_n(x)dx+int_a^pi f_n(x)dxright|le a+left|int_a^pi f_n(x)dxright|$$
Now, for all $epsilon>0$, let $a=epsilon$, and the remaining integral term tends to $0$ as $nto0$ (as $a$ is fixed as soon as $epsilon$ is fixed), so there is a $N$ such that for all $n>N$, $left|int_a^pi f_n(x)dxright|<epsilon$.
Hence, for all $epsilon>0$, there is a $N$ such that for all $n>N$, $left|int_0^pi f_n(x)dxright|<2epsilon$.
Hence $int_0^pi f_n(x)dxto0$ as $ntoinfty$.
answered Nov 23 at 21:15
Jean-Claude Arbaut
14.7k63464
For all $x$, $|f_n(x)|le1$, hence
$$left|int_0^{a}f_n(x)dxright|le (a-0) sup_{[0,a]}|f_n|le a$$
Therefore
$$left|int_0^pi f_n(x)dxright|=left|int_0^a f_n(x)dx+int_a^pi f_n(x)dxright|le a+left|int_a^pi f_n(x)dxright|$$
Now, for all $epsilon>0$, let $a=epsilon$, and the remaining integral term tends to $0$ as $nto0$ (as $a$ is fixed as soon as $epsilon$ is fixed), so there is a $N$ such that for all $n>N$, $left|int_a^pi f_n(x)dxright|<epsilon$.
Hence, for all $epsilon>0$, there is a $N$ such that for all $n>N$, $left|int_0^pi f_n(x)dxright|<2epsilon$.
Hence $int_0^pi f_n(x)dxto0$ as $ntoinfty$.
answered Nov 23 at 21:15
Jean-Claude Arbaut
14.7k63464
answered Nov 23 at 21:15
Jean-Claude Arbaut
14.7k63464
answered Nov 23 at 21:15
Jean-Claude Arbaut
14.7k63464
answered Nov 23 at 21:15
Jean-Claude Arbaut
14.7k63464
14.7k63464
Thank you ..I have a question....first the given sequences of function are not defined at 0 but since the limit exists at zero and the limiting value is 1 , I can define $ f_n(0) $=1, so did you write $ | f_n(x)| le 1$ with the same thought I have or something else ..If there is an other explanation please tell me ....so then $ f_n(x) $ is uniformly bounded and I can use bounded convergence theorem to conclude that the limiting value is zero..can I do that ?
– Normal
Nov 23 at 22:19
@Normal Yes, you can define $f_n(0)=1$ because the limit exists. It's a removable singularity, and the function thus defined is $C^{infty}$. See also en.wikipedia.org/wiki/Sinc_function
– Jean-Claude Arbaut
Nov 23 at 22:21
Thank you... please tell me it will be ok if I use bounded convergence theorem by using the uniform boundedness of $ f_n(x) $
– Normal
Nov 23 at 22:38
@Normal Yes, it also works. But you're not really a beginner then :)
– Jean-Claude Arbaut
Nov 23 at 22:41
One more question what if I choose an $ epsilon $ different from a .
– Normal
Nov 23 at 23:05
|
show 2 more comments
Thank you ..I have a question....first the given sequences of function are not defined at 0 but since the limit exists at zero and the limiting value is 1 , I can define $ f_n(0) $=1, so did you write $ | f_n(x)| le 1$ with the same thought I have or something else ..If there is an other explanation please tell me ....so then $ f_n(x) $ is uniformly bounded and I can use bounded convergence theorem to conclude that the limiting value is zero..can I do that ?
– Normal
Nov 23 at 22:19
@Normal Yes, you can define $f_n(0)=1$ because the limit exists. It's a removable singularity, and the function thus defined is $C^{infty}$. See also en.wikipedia.org/wiki/Sinc_function
– Jean-Claude Arbaut
Nov 23 at 22:21
Thank you... please tell me it will be ok if I use bounded convergence theorem by using the uniform boundedness of $ f_n(x) $
– Normal
Nov 23 at 22:38
@Normal Yes, it also works. But you're not really a beginner then :)
– Jean-Claude Arbaut
Nov 23 at 22:41
One more question what if I choose an $ epsilon $ different from a .
– Normal
Nov 23 at 23:05
Thank you ..I have a question....first the given sequences of function are not defined at 0 but since the limit exists at zero and the limiting value is 1 , I can define $ f_n(0) $=1, so did you write $ | f_n(x)| le 1$ with the same thought I have or something else ..If there is an other explanation please tell me ....so then $ f_n(x) $ is uniformly bounded and I can use bounded convergence theorem to conclude that the limiting value is zero..can I do that ?
– Normal
Nov 23 at 22:19
Thank you ..I have a question....first the given sequences of function are not defined at 0 but since the limit exists at zero and the limiting value is 1 , I can define $ f_n(0) $=1, so did you write $ | f_n(x)| le 1$ with the same thought I have or something else ..If there is an other explanation please tell me ....so then $ f_n(x) $ is uniformly bounded and I can use bounded convergence theorem to conclude that the limiting value is zero..can I do that ?
– Normal
Nov 23 at 22:19
@Normal Yes, you can define $f_n(0)=1$ because the limit exists. It's a removable singularity, and the function thus defined is $C^{infty}$. See also en.wikipedia.org/wiki/Sinc_function
– Jean-Claude Arbaut
Nov 23 at 22:21
@Normal Yes, you can define $f_n(0)=1$ because the limit exists. It's a removable singularity, and the function thus defined is $C^{infty}$. See also en.wikipedia.org/wiki/Sinc_function
– Jean-Claude Arbaut
Nov 23 at 22:21
Thank you... please tell me it will be ok if I use bounded convergence theorem by using the uniform boundedness of $ f_n(x) $
– Normal
Nov 23 at 22:38
Thank you... please tell me it will be ok if I use bounded convergence theorem by using the uniform boundedness of $ f_n(x) $
– Normal
Nov 23 at 22:38
@Normal Yes, it also works. But you're not really a beginner then :)
– Jean-Claude Arbaut
Nov 23 at 22:41
@Normal Yes, it also works. But you're not really a beginner then :)
– Jean-Claude Arbaut
Nov 23 at 22:41
One more question what if I choose an $ epsilon $ different from a .
– Normal
Nov 23 at 23:05
One more question what if I choose an $ epsilon $ different from a .
– Normal
Nov 23 at 23:05
|
show 2 more comments
We have that $$int_{0}^{pi }frac{sin nx}{nx} mathrm dx=int_{0}^{npi }frac{sin nx}{nx} mathrm {1over n}dnx={1over n }int_{0}^{npi} {sin xover x}dx$$also we know that $$int_0^{infty}{sin xover x}dx={pi over 2}$$according to Integration of Sinc function therefore $$lim _{nto infty}int_{0}^{pi }frac{sin nx}{nx} mathrm dx=0$$
answered Nov 23 at 19:43
Mostafa Ayaz
13.7k3836
Thank you .So $ lim_{nto infty} int_{0}^{pi} frac{sin nx}{nx} $=$( lim_{nto infty} {1 over n})(lim_{nto infty} int_{0}^{ npi}{sin x over x}) dx $ =0×(pi/2)=0. Is it what you did?
– Normal
Nov 23 at 20:05
You're welcome. Also could be explained that way. Yes...
– Mostafa Ayaz
Nov 23 at 20:10
Sorry for disturbing you again ,I have a doubt am I allowed to write what I wrote above..I mean I know lim ($ x_n y_n $ ) = lim ($ x_n $)lim ( $ y_n $) when the both limit exists ..but am I allowed to do the same thing in here.
– Normal
Nov 23 at 21:18
No mind at all :) Notice that i concluded that $$0le lim int_{0}^{pi }frac{sin nx}{nx} mathrm dx= lim {1over n }int_{0}^{npi} {sin xover x}dxlelim {1over n }int_{0}^{infty} {sin xover x}dx=lim {pi over 2n}=0$$ and used squeeze theorem
– Mostafa Ayaz
Nov 23 at 21:21
Thanks for answering though I didn't understand the line where you wrote lim $ {1 over n} int_{0}^{npi} { sin x over x} dx le lim {1 over n} int_{0}^{infty} {sin x over x} dx $ but it's alright the function is uniformly bounded defining $ f_n(0) =1 $ so by bounded convergence theorem I can conclude the answer is zero .
– Normal
Nov 23 at 22:31
|
show 1 more comment
We have that $$int_{0}^{pi }frac{sin nx}{nx} mathrm dx=int_{0}^{npi }frac{sin nx}{nx} mathrm {1over n}dnx={1over n }int_{0}^{npi} {sin xover x}dx$$also we know that $$int_0^{infty}{sin xover x}dx={pi over 2}$$according to Integration of Sinc function therefore $$lim _{nto infty}int_{0}^{pi }frac{sin nx}{nx} mathrm dx=0$$
answered Nov 23 at 19:43
Mostafa Ayaz
13.7k3836
Thank you .So $ lim_{nto infty} int_{0}^{pi} frac{sin nx}{nx} $=$( lim_{nto infty} {1 over n})(lim_{nto infty} int_{0}^{ npi}{sin x over x}) dx $ =0×(pi/2)=0. Is it what you did?
– Normal
Nov 23 at 20:05
You're welcome. Also could be explained that way. Yes...
– Mostafa Ayaz
Nov 23 at 20:10
Sorry for disturbing you again ,I have a doubt am I allowed to write what I wrote above..I mean I know lim ($ x_n y_n $ ) = lim ($ x_n $)lim ( $ y_n $) when the both limit exists ..but am I allowed to do the same thing in here.
– Normal
Nov 23 at 21:18
No mind at all :) Notice that i concluded that $$0le lim int_{0}^{pi }frac{sin nx}{nx} mathrm dx= lim {1over n }int_{0}^{npi} {sin xover x}dxlelim {1over n }int_{0}^{infty} {sin xover x}dx=lim {pi over 2n}=0$$ and used squeeze theorem
– Mostafa Ayaz
Nov 23 at 21:21
Thanks for answering though I didn't understand the line where you wrote lim $ {1 over n} int_{0}^{npi} { sin x over x} dx le lim {1 over n} int_{0}^{infty} {sin x over x} dx $ but it's alright the function is uniformly bounded defining $ f_n(0) =1 $ so by bounded convergence theorem I can conclude the answer is zero .
– Normal
Nov 23 at 22:31
|
show 1 more comment
We have that $$int_{0}^{pi }frac{sin nx}{nx} mathrm dx=int_{0}^{npi }frac{sin nx}{nx} mathrm {1over n}dnx={1over n }int_{0}^{npi} {sin xover x}dx$$also we know that $$int_0^{infty}{sin xover x}dx={pi over 2}$$according to Integration of Sinc function therefore $$lim _{nto infty}int_{0}^{pi }frac{sin nx}{nx} mathrm dx=0$$
answered Nov 23 at 19:43
Mostafa Ayaz
13.7k3836
We have that $$int_{0}^{pi }frac{sin nx}{nx} mathrm dx=int_{0}^{npi }frac{sin nx}{nx} mathrm {1over n}dnx={1over n }int_{0}^{npi} {sin xover x}dx$$also we know that $$int_0^{infty}{sin xover x}dx={pi over 2}$$according to Integration of Sinc function therefore $$lim _{nto infty}int_{0}^{pi }frac{sin nx}{nx} mathrm dx=0$$
answered Nov 23 at 19:43
Mostafa Ayaz
13.7k3836
answered Nov 23 at 19:43
Mostafa Ayaz
13.7k3836
answered Nov 23 at 19:43
Mostafa Ayaz
13.7k3836
answered Nov 23 at 19:43
Mostafa Ayaz
13.7k3836
13.7k3836
Thank you .So $ lim_{nto infty} int_{0}^{pi} frac{sin nx}{nx} $=$( lim_{nto infty} {1 over n})(lim_{nto infty} int_{0}^{ npi}{sin x over x}) dx $ =0×(pi/2)=0. Is it what you did?
– Normal
Nov 23 at 20:05
You're welcome. Also could be explained that way. Yes...
– Mostafa Ayaz
Nov 23 at 20:10
Sorry for disturbing you again ,I have a doubt am I allowed to write what I wrote above..I mean I know lim ($ x_n y_n $ ) = lim ($ x_n $)lim ( $ y_n $) when the both limit exists ..but am I allowed to do the same thing in here.
– Normal
Nov 23 at 21:18
No mind at all :) Notice that i concluded that $$0le lim int_{0}^{pi }frac{sin nx}{nx} mathrm dx= lim {1over n }int_{0}^{npi} {sin xover x}dxlelim {1over n }int_{0}^{infty} {sin xover x}dx=lim {pi over 2n}=0$$ and used squeeze theorem
– Mostafa Ayaz
Nov 23 at 21:21
Thanks for answering though I didn't understand the line where you wrote lim $ {1 over n} int_{0}^{npi} { sin x over x} dx le lim {1 over n} int_{0}^{infty} {sin x over x} dx $ but it's alright the function is uniformly bounded defining $ f_n(0) =1 $ so by bounded convergence theorem I can conclude the answer is zero .
– Normal
Nov 23 at 22:31
|
show 1 more comment
Thank you .So $ lim_{nto infty} int_{0}^{pi} frac{sin nx}{nx} $=$( lim_{nto infty} {1 over n})(lim_{nto infty} int_{0}^{ npi}{sin x over x}) dx $ =0×(pi/2)=0. Is it what you did?
– Normal
Nov 23 at 20:05
You're welcome. Also could be explained that way. Yes...
– Mostafa Ayaz
Nov 23 at 20:10
Sorry for disturbing you again ,I have a doubt am I allowed to write what I wrote above..I mean I know lim ($ x_n y_n $ ) = lim ($ x_n $)lim ( $ y_n $) when the both limit exists ..but am I allowed to do the same thing in here.
– Normal
Nov 23 at 21:18
No mind at all :) Notice that i concluded that $$0le lim int_{0}^{pi }frac{sin nx}{nx} mathrm dx= lim {1over n }int_{0}^{npi} {sin xover x}dxlelim {1over n }int_{0}^{infty} {sin xover x}dx=lim {pi over 2n}=0$$ and used squeeze theorem
– Mostafa Ayaz
Nov 23 at 21:21
Thanks for answering though I didn't understand the line where you wrote lim $ {1 over n} int_{0}^{npi} { sin x over x} dx le lim {1 over n} int_{0}^{infty} {sin x over x} dx $ but it's alright the function is uniformly bounded defining $ f_n(0) =1 $ so by bounded convergence theorem I can conclude the answer is zero .
– Normal
Nov 23 at 22:31
Thank you .So $ lim_{nto infty} int_{0}^{pi} frac{sin nx}{nx} $=$( lim_{nto infty} {1 over n})(lim_{nto infty} int_{0}^{ npi}{sin x over x}) dx $ =0×(pi/2)=0. Is it what you did?
– Normal
Nov 23 at 20:05
Thank you .So $ lim_{nto infty} int_{0}^{pi} frac{sin nx}{nx} $=$( lim_{nto infty} {1 over n})(lim_{nto infty} int_{0}^{ npi}{sin x over x}) dx $ =0×(pi/2)=0. Is it what you did?
– Normal
Nov 23 at 20:05
You're welcome. Also could be explained that way. Yes...
– Mostafa Ayaz
Nov 23 at 20:10
You're welcome. Also could be explained that way. Yes...
– Mostafa Ayaz
Nov 23 at 20:10
Sorry for disturbing you again ,I have a doubt am I allowed to write what I wrote above..I mean I know lim ($ x_n y_n $ ) = lim ($ x_n $)lim ( $ y_n $) when the both limit exists ..but am I allowed to do the same thing in here.
– Normal
Nov 23 at 21:18
Sorry for disturbing you again ,I have a doubt am I allowed to write what I wrote above..I mean I know lim ($ x_n y_n $ ) = lim ($ x_n $)lim ( $ y_n $) when the both limit exists ..but am I allowed to do the same thing in here.
– Normal
Nov 23 at 21:18
No mind at all :) Notice that i concluded that $$0le lim int_{0}^{pi }frac{sin nx}{nx} mathrm dx= lim {1over n }int_{0}^{npi} {sin xover x}dxlelim {1over n }int_{0}^{infty} {sin xover x}dx=lim {pi over 2n}=0$$ and used squeeze theorem
– Mostafa Ayaz
Nov 23 at 21:21
No mind at all :) Notice that i concluded that $$0le lim int_{0}^{pi }frac{sin nx}{nx} mathrm dx= lim {1over n }int_{0}^{npi} {sin xover x}dxlelim {1over n }int_{0}^{infty} {sin xover x}dx=lim {pi over 2n}=0$$ and used squeeze theorem
– Mostafa Ayaz
Nov 23 at 21:21
Thanks for answering though I didn't understand the line where you wrote lim $ {1 over n} int_{0}^{npi} { sin x over x} dx le lim {1 over n} int_{0}^{infty} {sin x over x} dx $ but it's alright the function is uniformly bounded defining $ f_n(0) =1 $ so by bounded convergence theorem I can conclude the answer is zero .
– Normal
Nov 23 at 22:31
Thanks for answering though I didn't understand the line where you wrote lim $ {1 over n} int_{0}^{npi} { sin x over x} dx le lim {1 over n} int_{0}^{infty} {sin x over x} dx $ but it's alright the function is uniformly bounded defining $ f_n(0) =1 $ so by bounded convergence theorem I can conclude the answer is zero .
– Normal
Nov 23 at 22:31
|
show 1 more comment
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is this what you aimed to write? Please check.
– user376343
Nov 23 at 19:34
1
Where is $a$ in the integral?
– Will M.
Nov 23 at 19:34
No, sorry, I fixed it
– Normal
Nov 23 at 19:38
The integrand is continuous and bounded, so what happens when you integrate on $[0,a]$ with $ato0$?
– Jean-Claude Arbaut
Nov 23 at 19:45
The integrand is continuous on [0,pi] considering $ f_n(0) $=1 and hence it is bounded on [0,pi]. Sorry I am a beginner in real analysis ..I could not understand what you said .
– Normal
Nov 23 at 20:17