Prove that the function $f(x) = sum_{k =1}^{infty} frac{sin(kx)}{2^{k}}$ is infinitely differentiable
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
1696
add a comment |
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
1696
I think you could just justify the operation of differentiation term by term. Hint: consider the uniform convergence of such series.
– xbh
Dec 12 '18 at 6:58
See math.stackexchange.com/questions/2446315/…
– lab bhattacharjee
Dec 12 '18 at 7:00
Think of $x$ as $z$ a complex variable. Then ask if $f(z)$ is analytic.
– R.C.Cowsik
Dec 12 '18 at 9:25
add a comment |
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
1696
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
1696
asked Dec 12 '18 at 6:54
stackofhay42stackofhay42
1696
asked Dec 12 '18 at 6:54
stackofhay42stackofhay42
1696
asked Dec 12 '18 at 6:54
stackofhay42stackofhay42
1696
asked Dec 12 '18 at 6:54
stackofhay42stackofhay42
1696
1696
I think you could just justify the operation of differentiation term by term. Hint: consider the uniform convergence of such series.
– xbh
Dec 12 '18 at 6:58
See math.stackexchange.com/questions/2446315/…
– lab bhattacharjee
Dec 12 '18 at 7:00
Think of $x$ as $z$ a complex variable. Then ask if $f(z)$ is analytic.
– R.C.Cowsik
Dec 12 '18 at 9:25
add a comment |
I think you could just justify the operation of differentiation term by term. Hint: consider the uniform convergence of such series.
– xbh
Dec 12 '18 at 6:58
See math.stackexchange.com/questions/2446315/…
– lab bhattacharjee
Dec 12 '18 at 7:00
Think of $x$ as $z$ a complex variable. Then ask if $f(z)$ is analytic.
– R.C.Cowsik
Dec 12 '18 at 9:25
I think you could just justify the operation of differentiation term by term. Hint: consider the uniform convergence of such series.
– xbh
Dec 12 '18 at 6:58
I think you could just justify the operation of differentiation term by term. Hint: consider the uniform convergence of such series.
– xbh
Dec 12 '18 at 6:58
See math.stackexchange.com/questions/2446315/…
– lab bhattacharjee
Dec 12 '18 at 7:00
See math.stackexchange.com/questions/2446315/…
– lab bhattacharjee
Dec 12 '18 at 7:00
Think of $x$ as $z$ a complex variable. Then ask if $f(z)$ is analytic.
– R.C.Cowsik
Dec 12 '18 at 9:25
Think of $x$ as $z$ a complex variable. Then ask if $f(z)$ is analytic.
– R.C.Cowsik
Dec 12 '18 at 9:25
add a comment |
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I think you could just justify the operation of differentiation term by term. Hint: consider the uniform convergence of such series.
– xbh
Dec 12 '18 at 6:58
See math.stackexchange.com/questions/2446315/…
– lab bhattacharjee
Dec 12 '18 at 7:00
Think of $x$ as $z$ a complex variable. Then ask if $f(z)$ is analytic.
– R.C.Cowsik
Dec 12 '18 at 9:25