Fourier Coefficients & Integrals: An Analysis

[buzzmath]

Homework Statement

1. f is a function defined on the interval -a<x<a and has Fourier coefficients an=0 bn=1/n^(1/2) what can you say about the integral from -a to a of f^2(x)dx?
2. Show that as n goes to infinity the Fourier sine coefficients of the function f(x)=1/x -pi<x<pi tend to a nonzero constant. Use the fact that the integral from 0 to infinity of sin(t)/t dt=pi/2

Homework Equations

The Attempt at a Solution

1. Persevals equality says 1/a * integral from -a to a of f^2(x)dx = 2a0^2 + susm n=1 to infinity of an^2 + bn^2 so I was thinking that since to a's are zero that we could just say that f^2(x) converges to the sum of the b's but the book says the answer is f^2(x) doesn't converge since sum n=1 to infinity of (|an|+|bn|) is infinity but the a's are zero and bn = 1/n^(1/2) I thought these bn's would go to zero not infinity. What am I doing wrong?

2. f I was just going to compute the bn's from the integral directly and show that as n goes to infinity they go to a nonzero constant but I'm not sure how I would do this problem with the integral given. How does an integral from 0 to infinity come into this problem?

thanks

 

[quasar987]

1. The sequence 1/n goes to 0 as n goes to infinity, but the series $\sum 1/n$ does not!

2. Consider doing a change of coordinate.

 

[buzzmath]

I know that the series isn't zero but they say that the series goes to infinity which I don't understand since the sequence goes to zero. I mean how would they come up with this? is there something I'm not seeing?
What do you mean by doing a change of coordinates? how would I get integral from 0 to infinity?

 

[quasar987]

1. That the general term of a series goes to 0 is not a sufficient condition to assure that the series converge. The standard example used to illustrate this fact is precisely the series of 1/n, called the harmonic series: http://en.wikipedia.org/wiki/Harmonic_series_(mathematics)#Divergence_of_the_harmonic_series

2. Maybe you don't see where this is going yet, but you got to try something. A logical first step seems to me to try to get the integrand in the same form as the integral in the hint. In the integralof the Fourier coefficient, you have sin(nx)/x... you'd like to get sin(t)/t. What change of variables do you suggest? And what does that leave you with?