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Homework Statement
Prove Ʃ1/(n^2) as n goes to infinity = (∏^2)/8
Homework Equations
The Attempt at a Solution
No idea how to start. Pls guide.
Thanks
Homework Statement
Prove Ʃ1/(n^2) as n goes to infinity = (∏^2)/8
Homework Equations
The Attempt at a Solution
No idea how to start. Pls guide.
Thanks
Homework Statement
Prove Ʃ1/(n^2) as n goes to infinity = (∏^2)/8
Homework Equations
The Attempt at a Solution
No idea how to start. Pls guide.
Thanks
You need to show your work, first. Anyway, what is the context? Is this a problem in a course? If so, what is the course subject (analytic function theory, differential equations, Fourier analysis...)?
You won't be able to prove it, since the result is wrong. The sum converges to [itex]\frac{\pi^{2}}{6}[/itex] instead
Homework Statement
Prove Ʃ1/(n^2) as n goes to infinity = (∏^2)/8
Homework Equations
The Attempt at a Solution
No idea how to start. Pls guide.
Thanks
I got that from a manual solution, I don't know where should I ask my question so I put it here .
This answer is for quantum mechanics problem. I don't think that is Fourier series.
Sorry for my mistakes. Thanks for replying.
Do I need to know f(x)= ? for solving this problem? Like f(x)=x for showing 1 - (1/3) + (1/5)....= ∏/4 .as shown below
Use the same function, but instead of evaluating the series at a point, use Parseval's identity.
f(x)= x .Range? -∏ <x< ∏ ?
What do you mean by evaluating the series at a point? Fourier series evaluating at a point
Complex fouries series and Parseval's theorem ?
Why do we need two types?
Yes, exactly the same function, exactly the same series. Did you look up Parseval's theorem? Evaluating the series gives you things like 1/n, Parseval's gives you things like 1/n^2. Just try it.
Use the same function, but instead of evaluating the series at a point, use Parseval's identity.
I got the answer by using parseval's theorem when f(x)=x over the interval -∏<x<∏.
Thanks
But when do I know I need to use complex Fourier series and parseval's theorem? Is that because there is a square? (2m+1)^2
[tex] \sum_{n=0}^{\infty} \frac{1}{(2n+1)^2} = \frac{\pi^2}{8}.[/tex]
Can I say Fourier series is used when my conjugate ,g(x) is not complex. Complex conjugate of g(x) = g(x), Eg. ∫g(x)f(x)dx= c
But when g(x) is complex , complex conjugate of g(x) is not equal to g(x),then I should use complex Fourier series?
This is my own conclusion after reading . Please Correct if it's wrong.
I am still don't understand what do you mean evaluate the series at a point, is that meant f(x)= x , the x should not be a point, so I have to integrate by using parseval's , but how do you know we can't treat it as a point?
I got the answer by using parseval's theorem when f(x)=x over the interval -∏<x<∏.
Thanks
But when do I know I need to use complex Fourier series and parseval's theorem? Is that because there is a square? (2m+1)^2
[tex] \sum_{n=0}^{\infty} \frac{1}{(2n+1)^2} = \frac{\pi^2}{8}.[/tex]
Can I say Fourier series is used when my conjugate ,g(x) is not complex. Complex conjugate of g(x) = g(x), Eg. ∫g(x)f(x)dx= c
But when g(x) is complex , complex conjugate of g(x) is not equal to g(x),then I should use complex Fourier series?
This is my own conclusion after reading . Please Correct if it's wrong.
I am still don't understand what do you mean evaluate the series at a point, is that meant f(x)= x , the x should not be a point, so I have to integrate by using parseval's , but how do you know we can't treat it as a point?
Yes, the square is a hint you might need Parseval. But you didn't really finish the proof you presented in post 8. Could you do that? Then you might see why evaluation of the Fourier series at a point can give you the sum of some series.