Finding the sum of an infinite series using Fourier

In summary, the homework statement is trying to find the sum of (-1)3n+1/(2n-1)3. by using term-by-term integration on the cosine Fourier series x= L/2-4L/π2∑cos(((2n-1)πx)/L)/(2n-1)2. The Attempt at a Solution is when integrating and substituting Lx/2 for x's sine Fourier series I get ∑L2/π(2n-1)[(-1)n+1-4/π2(2n-1)2]sin(((2n-1)πx)/L) for the final
  • #1
John Jacke
12
0

Homework Statement


Trying to find the sum of (-1)3n+1/(2n-1)3. by using term-by-term integration on the cosine Fourier series x= L/2-4L/π2∑cos(((2n-1)πx)/L)/(2n-1)2.

Homework Equations


Shown below

The Attempt at a Solution


When integrating and substituting Lx/2 for x's sine Fourier series I get ∑L2/π(2n-1)[(-1)n+1-4/π2(2n-1)2]sin(((2n-1)πx)/L) for the final series.To find the sum of the series in class we would find Bn explicitly using the formula and then set it equal to this coefficient found by term by term integration. However when I do that they are the same except for that the only difference is that one term alternates and one does not in sign. So explicitly solving for Bn gives me what you see above except the first term doesn't alternate. So setting them equal cancels everything except those terms. Is this supposed to mean anything? I've gotten the correct answer but was off by a factor of 2. It should (I think) be π3/32.
 
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  • #2
One thing that I believe you are missing is the original ## f(x) ##, which is the periodic function that generates this series. Just a hint=I think I have this correct=is that ## f(x) ## is an even triangle shaped function. You need to integrate this ## f(x) ## on the left side of the equation. ( Unlike your post where you call it ## x ##, it s ## f(x) ##) . And also when you integrate the term ## L/2 ## from 0 to ## L/2 ##, you will get ## L^2/4 ##. ## \\ ## Editing: This one has a trick to it in the final integral that is used to evaluate the series=you would normally expect to evaluate this integral over the interval ## -L/2 ## to ## +L/2 ## or from ## 0 ## to ## L/2 ##. Try an ## L/4 ## limit instead. Also, in giving it in the functional form with an ## L ## in the amplitude, they could easily call this "1"" instead, and the same goes for the x limits on the triangle waveform. The waveform can go from ## -1/2 ## to ## +1/2 ## simplifying the algebra. (Additional editing, just so that I don't steer you in the wrong direction: I believe the function they gave you has a period equal to ##T=2L ##, but I'll let you verify that when you analyze the Fourier series for it.) ## \\ ## Editing: It is possible to use the waveform in the form ## y=x ## but that is only half of the even triangle. The other half ## y=-x ## needs to be included. In this case you could let the period go from -1 to 1. There are a couple of ways of choosing the even function triangle, but ## y=x ## by itself does not work. Looking over the form that they used, it looks like they may have ## f(0)=0 ## and ## f(x) ## increases as x increases. The identity can be proven with any of a couple of even triangle-shaped functions.(I succeeded at proving it, and I'm still working on figuring out the precise function that they supplied=additional edit=I figured out now, but I'll let you work out that part yourself.) It takes a little extra effort to figure out what the precise triangle is that they have for the function they gave you. It sort of works by trial and error. You try an even triangle function and compute its Fourier components and see if you have a match...Editing... Alternatively, a good computer graphics package would quickly graph out the function they gave you, but I don't have that resource presently available. ## \\ ## One additional suggestion: they could step you through the final integral instead of getting a zero=zero result if you integrate over the whole period of the function.
 
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1. What is an infinite series?

An infinite series is a sum of an infinite number of terms. It is represented by the following notation: ∑n=1∞ an, where an is the nth term of the series.

2. How is Fourier used to find the sum of an infinite series?

Fourier analysis is a mathematical technique that can be used to find the sum of certain types of infinite series. It involves breaking down a function into its component frequencies and representing it as a sum of sine and cosine waves. This allows us to find the coefficients of the series and ultimately determine its sum.

3. What types of series can be solved using Fourier?

Fourier can be used to solve series that follow specific patterns, such as geometric series or those that can be represented as trigonometric functions. It is not applicable to all infinite series.

4. What are the steps for finding the sum of an infinite series using Fourier?

The steps for solving an infinite series using Fourier are as follows: 1) Identify the type of series and determine if it can be solved using Fourier. 2) Use Fourier analysis to break down the function into its component frequencies. 3) Find the coefficients of the series by solving for the Fourier coefficients. 4) Use the coefficients to determine the sum of the series.

5. Can Fourier be used to find the sum of any series?

No, Fourier is only applicable to certain types of series that follow specific patterns. It cannot be used to find the sum of all infinite series.

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