Fourier series periodic function

[dpackard]

Homework Statement

The following specification of coefficients defines a Fourier series:
$$a_{0}=\frac{2}{\pi};&a_{1}=\frac{1}{2};&a_{n}=\left\{\begin{array}{cc}-\frac{2}{\pi}(-1)^\frac{n}{2}\frac{1}{n^2-1}&\mbox{ for }n\mbox{ even }\\0&\mbox{ for }n\mbox{ odd}\end{array}\right.(\mbox{for }n\geq2); \\ \mbox{and} \\b_n=0\mbox{ for all }n$$

Homework Equations

$$f(x)=a_0+\sum a_n\cos(nx)+\sum b_n\sin(nx)$$

The Attempt at a Solution

I graphed this up to n=4 and it resembles semi-circles on top of a line, and I'm supposed to figure out what "particular periodic function this is. Anyone know?

 

[mighty2000]

it is important to approach this problem by first understanding the given coefficients and how they relate to the Fourier series equation. From the given specification, we can see that the cosines and sines have alternating coefficients, with a_n being non-zero only for even n values and b_n being zero for all n values. This means that the resulting function will have a periodicity of 2π, as the cosine term will only contribute to the even harmonics and the sine term will not contribute at all.

Next, we can examine the specific values of the coefficients. The value of a_0 is 2/π, which is the average value of the function over one period. This indicates that the function will have a constant value of 2/π over the entire period.

For a_n, we can see that the value for even n will be non-zero and oscillate between positive and negative values. This suggests that the function will have a periodic behavior that alternates between positive and negative values. The amplitude of this oscillation decreases as n increases, as the denominator (n^2-1) will increase faster than the numerator (-1)^n, resulting in a smaller overall value.

Putting this all together, we can see that the resulting function will be a periodic function with a constant value of 2/π and oscillations between positive and negative values with decreasing amplitude. This is similar to a square wave, but instead of sharp corners, the function will have smoother transitions due to the cosine term.

Therefore, the particular periodic function represented by this Fourier series is a smoothed out version of a square wave with a period of 2π and an average value of 2/π.