Prove that the function $f(x) = sum_{k =1}^{infty} frac{sin(kx)}{2^{k}}$ is infinitely differentiable
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Prove that the function $f(x) = sum_{k =1}^{infty}
frac{sin(kx)}{2^{k}}$ is infinitely differentiable
This is a practice problem for an exam I have coming soon. I am trying to study, but I cannot figure out how to do this problem. I think that the correct approach would be to show that the function $f$ is analytic, which implies that $f$ is infinitely differentiable.
I'm not sure if this is the correct approach, and how to do it, though.
Any help is appreciated.
real-analysis analysis derivatives power-series taylor-expansion
asked Dec 12 '18 at 6:54
stackofhay42stackofhay42
1956
$endgroup$
add a comment |
$begingroup$
Prove that the function $f(x) = sum_{k =1}^{infty}
frac{sin(kx)}{2^{k}}$ is infinitely differentiable
This is a practice problem for an exam I have coming soon. I am trying to study, but I cannot figure out how to do this problem. I think that the correct approach would be to show that the function $f$ is analytic, which implies that $f$ is infinitely differentiable.
I'm not sure if this is the correct approach, and how to do it, though.
Any help is appreciated.
real-analysis analysis derivatives power-series taylor-expansion
asked Dec 12 '18 at 6:54
stackofhay42stackofhay42
1956
$endgroup$
$begingroup$
I think you could just justify the operation of differentiation term by term. Hint: consider the uniform convergence of such series.
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– xbh
Dec 12 '18 at 6:58
$begingroup$
See math.stackexchange.com/questions/2446315/…
$endgroup$
– lab bhattacharjee
Dec 12 '18 at 7:00
$begingroup$
Think of $x$ as $z$ a complex variable. Then ask if $f(z)$ is analytic.
$endgroup$
– R.C.Cowsik
Dec 12 '18 at 9:25
add a comment |
$begingroup$
Prove that the function $f(x) = sum_{k =1}^{infty}
frac{sin(kx)}{2^{k}}$ is infinitely differentiable
This is a practice problem for an exam I have coming soon. I am trying to study, but I cannot figure out how to do this problem. I think that the correct approach would be to show that the function $f$ is analytic, which implies that $f$ is infinitely differentiable.
I'm not sure if this is the correct approach, and how to do it, though.
Any help is appreciated.
real-analysis analysis derivatives power-series taylor-expansion
asked Dec 12 '18 at 6:54
stackofhay42stackofhay42
1956
$endgroup$
Prove that the function $f(x) = sum_{k =1}^{infty}
frac{sin(kx)}{2^{k}}$ is infinitely differentiable
This is a practice problem for an exam I have coming soon. I am trying to study, but I cannot figure out how to do this problem. I think that the correct approach would be to show that the function $f$ is analytic, which implies that $f$ is infinitely differentiable.
I'm not sure if this is the correct approach, and how to do it, though.
Any help is appreciated.
real-analysis analysis derivatives power-series taylor-expansion
real-analysis analysis derivatives power-series taylor-expansion
asked Dec 12 '18 at 6:54
stackofhay42stackofhay42
1956
asked Dec 12 '18 at 6:54
stackofhay42stackofhay42
1956
asked Dec 12 '18 at 6:54
stackofhay42stackofhay42
1956
asked Dec 12 '18 at 6:54
stackofhay42stackofhay42
1956
asked Dec 12 '18 at 6:54
stackofhay42stackofhay42
1956
1956
$begingroup$
I think you could just justify the operation of differentiation term by term. Hint: consider the uniform convergence of such series.
$endgroup$
– xbh
Dec 12 '18 at 6:58
$begingroup$
See math.stackexchange.com/questions/2446315/…
$endgroup$
– lab bhattacharjee
Dec 12 '18 at 7:00
$begingroup$
Think of $x$ as $z$ a complex variable. Then ask if $f(z)$ is analytic.
$endgroup$
– R.C.Cowsik
Dec 12 '18 at 9:25
add a comment |
$begingroup$
I think you could just justify the operation of differentiation term by term. Hint: consider the uniform convergence of such series.
$endgroup$
– xbh
Dec 12 '18 at 6:58
$begingroup$
See math.stackexchange.com/questions/2446315/…
$endgroup$
– lab bhattacharjee
Dec 12 '18 at 7:00
$begingroup$
Think of $x$ as $z$ a complex variable. Then ask if $f(z)$ is analytic.
$endgroup$
– R.C.Cowsik
Dec 12 '18 at 9:25
$begingroup$
I think you could just justify the operation of differentiation term by term. Hint: consider the uniform convergence of such series.
$endgroup$
– xbh
Dec 12 '18 at 6:58
$begingroup$
I think you could just justify the operation of differentiation term by term. Hint: consider the uniform convergence of such series.
$endgroup$
– xbh
Dec 12 '18 at 6:58
$begingroup$
See math.stackexchange.com/questions/2446315/…
$endgroup$
– lab bhattacharjee
Dec 12 '18 at 7:00
$begingroup$
See math.stackexchange.com/questions/2446315/…
$endgroup$
– lab bhattacharjee
Dec 12 '18 at 7:00
$begingroup$
Think of $x$ as $z$ a complex variable. Then ask if $f(z)$ is analytic.
$endgroup$
– R.C.Cowsik
Dec 12 '18 at 9:25
$begingroup$
Think of $x$ as $z$ a complex variable. Then ask if $f(z)$ is analytic.
$endgroup$
– R.C.Cowsik
Dec 12 '18 at 9:25
add a comment |
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$begingroup$
I think you could just justify the operation of differentiation term by term. Hint: consider the uniform convergence of such series.
$endgroup$
– xbh
Dec 12 '18 at 6:58
$begingroup$
See math.stackexchange.com/questions/2446315/…
$endgroup$
– lab bhattacharjee
Dec 12 '18 at 7:00
$begingroup$
Think of $x$ as $z$ a complex variable. Then ask if $f(z)$ is analytic.
$endgroup$
– R.C.Cowsik
Dec 12 '18 at 9:25