Testing the convergency of a sequence.
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Question. Let $~fin C^1[-pi,pi]$ be such that $f(-pi)=f(pi)$. Define $$a_n=int_-pi^pif(t)cos nt~dt~, ~nin mathbbN$$
Which of the following statements are true?
a. The sequence $a_n$ is bounded.
b. The sequence $na_n$ converges to zero as $n to infty$
c. The series $sum_n=1^inftyn^2|a_n|^2$ is convergent.
My Solution.
(a) True. $|a_n|le int_-pi^pi|f(t)||cos nt|dt le int_-pi^pi|f(t)|dt, ~ textfor all ~n in mathbbN$.
But I can't prove/disprove the other two options. Please help..thank you.
real-analysis sequences-and-series
asked Aug 13 at 8:59
Indrajit Ghosh
657415
add a comment |Â
up vote
1
down vote
favorite
Question. Let $~fin C^1[-pi,pi]$ be such that $f(-pi)=f(pi)$. Define $$a_n=int_-pi^pif(t)cos nt~dt~, ~nin mathbbN$$
Which of the following statements are true?
a. The sequence $a_n$ is bounded.
b. The sequence $na_n$ converges to zero as $n to infty$
c. The series $sum_n=1^inftyn^2|a_n|^2$ is convergent.
My Solution.
(a) True. $|a_n|le int_-pi^pi|f(t)||cos nt|dt le int_-pi^pi|f(t)|dt, ~ textfor all ~n in mathbbN$.
But I can't prove/disprove the other two options. Please help..thank you.
real-analysis sequences-and-series
asked Aug 13 at 8:59
Indrajit Ghosh
657415
$f'in C^0(-pi,pi)$ so $f'in L^2(-pi,pi)$
– Jack D'Aurizio♦
Aug 13 at 9:22
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Question. Let $~fin C^1[-pi,pi]$ be such that $f(-pi)=f(pi)$. Define $$a_n=int_-pi^pif(t)cos nt~dt~, ~nin mathbbN$$
Which of the following statements are true?
a. The sequence $a_n$ is bounded.
b. The sequence $na_n$ converges to zero as $n to infty$
c. The series $sum_n=1^inftyn^2|a_n|^2$ is convergent.
My Solution.
(a) True. $|a_n|le int_-pi^pi|f(t)||cos nt|dt le int_-pi^pi|f(t)|dt, ~ textfor all ~n in mathbbN$.
But I can't prove/disprove the other two options. Please help..thank you.
real-analysis sequences-and-series
asked Aug 13 at 8:59
Indrajit Ghosh
657415
Question. Let $~fin C^1[-pi,pi]$ be such that $f(-pi)=f(pi)$. Define $$a_n=int_-pi^pif(t)cos nt~dt~, ~nin mathbbN$$
Which of the following statements are true?
a. The sequence $a_n$ is bounded.
b. The sequence $na_n$ converges to zero as $n to infty$
c. The series $sum_n=1^inftyn^2|a_n|^2$ is convergent.
My Solution.
(a) True. $|a_n|le int_-pi^pi|f(t)||cos nt|dt le int_-pi^pi|f(t)|dt, ~ textfor all ~n in mathbbN$.
But I can't prove/disprove the other two options. Please help..thank you.
real-analysis sequences-and-series
asked Aug 13 at 8:59
Indrajit Ghosh
657415
asked Aug 13 at 8:59
Indrajit Ghosh
657415
asked Aug 13 at 8:59
Indrajit Ghosh
657415
asked Aug 13 at 8:59
Indrajit Ghosh
657415
657415
$f'in C^0(-pi,pi)$ so $f'in L^2(-pi,pi)$
– Jack D'Aurizio♦
Aug 13 at 9:22
add a comment |Â
$f'in C^0(-pi,pi)$ so $f'in L^2(-pi,pi)$
– Jack D'Aurizio♦
Aug 13 at 9:22
$f'in C^0(-pi,pi)$ so $f'in L^2(-pi,pi)$
– Jack D'Aurizio♦
Aug 13 at 9:22
$f'in C^0(-pi,pi)$ so $f'in L^2(-pi,pi)$
– Jack D'Aurizio♦
Aug 13 at 9:22
add a comment |Â
1 Answer
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For b) and c) you just have to integrate by parts. $a_n=-frac 1 n int_-pi ^pi f'(t) sin (nt) , dt$ so $na_n$ are the coefficients of the sine series of $g=-f'$ which is continuous and periodic. By standard results in the theory of Fourier series b) and c) are both true. In fact $pi sum n^2|a_n|^2leqint |f'(t)|^2 , dt $
answered Aug 13 at 9:16
Kavi Rama Murthy
22.2k2933
...sir can u please refer those results in Fourier series that u are talking...
– Indrajit Ghosh
Aug 13 at 9:19
Since c. implies b., there is need for only one standard result.
– uniquesolution
Aug 13 at 9:22
Almost any book on Fourier series has the result I have used. Examples are Apostol, (baby) Rudin, Fourier series by Edwards etc.
– Kavi Rama Murthy
Aug 13 at 9:22
@uniquesolution Very true. Parseval's identity is the only result needed.
– Kavi Rama Murthy
Aug 13 at 9:23
Thank you so much sir..@KaviRamaMurthy
– Indrajit Ghosh
Aug 13 at 9:23
 |Â
show 2 more comments
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
For b) and c) you just have to integrate by parts. $a_n=-frac 1 n int_-pi ^pi f'(t) sin (nt) , dt$ so $na_n$ are the coefficients of the sine series of $g=-f'$ which is continuous and periodic. By standard results in the theory of Fourier series b) and c) are both true. In fact $pi sum n^2|a_n|^2leqint |f'(t)|^2 , dt $
answered Aug 13 at 9:16
Kavi Rama Murthy
22.2k2933
...sir can u please refer those results in Fourier series that u are talking...
– Indrajit Ghosh
Aug 13 at 9:19
Since c. implies b., there is need for only one standard result.
– uniquesolution
Aug 13 at 9:22
Almost any book on Fourier series has the result I have used. Examples are Apostol, (baby) Rudin, Fourier series by Edwards etc.
– Kavi Rama Murthy
Aug 13 at 9:22
@uniquesolution Very true. Parseval's identity is the only result needed.
– Kavi Rama Murthy
Aug 13 at 9:23
Thank you so much sir..@KaviRamaMurthy
– Indrajit Ghosh
Aug 13 at 9:23
 |Â
show 2 more comments
up vote
2
down vote
accepted
For b) and c) you just have to integrate by parts. $a_n=-frac 1 n int_-pi ^pi f'(t) sin (nt) , dt$ so $na_n$ are the coefficients of the sine series of $g=-f'$ which is continuous and periodic. By standard results in the theory of Fourier series b) and c) are both true. In fact $pi sum n^2|a_n|^2leqint |f'(t)|^2 , dt $
answered Aug 13 at 9:16
Kavi Rama Murthy
22.2k2933
...sir can u please refer those results in Fourier series that u are talking...
– Indrajit Ghosh
Aug 13 at 9:19
Since c. implies b., there is need for only one standard result.
– uniquesolution
Aug 13 at 9:22
Almost any book on Fourier series has the result I have used. Examples are Apostol, (baby) Rudin, Fourier series by Edwards etc.
– Kavi Rama Murthy
Aug 13 at 9:22
@uniquesolution Very true. Parseval's identity is the only result needed.
– Kavi Rama Murthy
Aug 13 at 9:23
Thank you so much sir..@KaviRamaMurthy
– Indrajit Ghosh
Aug 13 at 9:23
 |Â
show 2 more comments
up vote
2
down vote
accepted
up vote
2
down vote
accepted
For b) and c) you just have to integrate by parts. $a_n=-frac 1 n int_-pi ^pi f'(t) sin (nt) , dt$ so $na_n$ are the coefficients of the sine series of $g=-f'$ which is continuous and periodic. By standard results in the theory of Fourier series b) and c) are both true. In fact $pi sum n^2|a_n|^2leqint |f'(t)|^2 , dt $
answered Aug 13 at 9:16
Kavi Rama Murthy
22.2k2933
For b) and c) you just have to integrate by parts. $a_n=-frac 1 n int_-pi ^pi f'(t) sin (nt) , dt$ so $na_n$ are the coefficients of the sine series of $g=-f'$ which is continuous and periodic. By standard results in the theory of Fourier series b) and c) are both true. In fact $pi sum n^2|a_n|^2leqint |f'(t)|^2 , dt $
answered Aug 13 at 9:16
Kavi Rama Murthy
22.2k2933
edited Aug 13 at 23:27
answered Aug 13 at 9:16
Kavi Rama Murthy
22.2k2933
answered Aug 13 at 9:16
Kavi Rama Murthy
22.2k2933
answered Aug 13 at 9:16
Kavi Rama Murthy
22.2k2933
22.2k2933
...sir can u please refer those results in Fourier series that u are talking...
– Indrajit Ghosh
Aug 13 at 9:19
Since c. implies b., there is need for only one standard result.
– uniquesolution
Aug 13 at 9:22
Almost any book on Fourier series has the result I have used. Examples are Apostol, (baby) Rudin, Fourier series by Edwards etc.
– Kavi Rama Murthy
Aug 13 at 9:22
@uniquesolution Very true. Parseval's identity is the only result needed.
– Kavi Rama Murthy
Aug 13 at 9:23
Thank you so much sir..@KaviRamaMurthy
– Indrajit Ghosh
Aug 13 at 9:23
 |Â
show 2 more comments
...sir can u please refer those results in Fourier series that u are talking...
– Indrajit Ghosh
Aug 13 at 9:19
Since c. implies b., there is need for only one standard result.
– uniquesolution
Aug 13 at 9:22
Almost any book on Fourier series has the result I have used. Examples are Apostol, (baby) Rudin, Fourier series by Edwards etc.
– Kavi Rama Murthy
Aug 13 at 9:22
@uniquesolution Very true. Parseval's identity is the only result needed.
– Kavi Rama Murthy
Aug 13 at 9:23
Thank you so much sir..@KaviRamaMurthy
– Indrajit Ghosh
Aug 13 at 9:23
...sir can u please refer those results in Fourier series that u are talking...
– Indrajit Ghosh
Aug 13 at 9:19
...sir can u please refer those results in Fourier series that u are talking...
– Indrajit Ghosh
Aug 13 at 9:19
Since c. implies b., there is need for only one standard result.
– uniquesolution
Aug 13 at 9:22
Since c. implies b., there is need for only one standard result.
– uniquesolution
Aug 13 at 9:22
Almost any book on Fourier series has the result I have used. Examples are Apostol, (baby) Rudin, Fourier series by Edwards etc.
– Kavi Rama Murthy
Aug 13 at 9:22
Almost any book on Fourier series has the result I have used. Examples are Apostol, (baby) Rudin, Fourier series by Edwards etc.
– Kavi Rama Murthy
Aug 13 at 9:22
@uniquesolution Very true. Parseval's identity is the only result needed.
– Kavi Rama Murthy
Aug 13 at 9:23
@uniquesolution Very true. Parseval's identity is the only result needed.
– Kavi Rama Murthy
Aug 13 at 9:23
Thank you so much sir..@KaviRamaMurthy
– Indrajit Ghosh
Aug 13 at 9:23
Thank you so much sir..@KaviRamaMurthy
– Indrajit Ghosh
Aug 13 at 9:23
 |Â
show 2 more comments
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$f'in C^0(-pi,pi)$ so $f'in L^2(-pi,pi)$
– Jack D'Aurizio♦
Aug 13 at 9:22