I need to sketch the graph to which this series converges:
...and I don't have the foggiest idea of how to approach this. Can someone give me an idea of how to go about this?
After reading your post, I took a quick look, but did not come across any series that matched this one.
Regardless, unless I made a mistake, the only relevant part of the question is pretty much worded as I stated it in my original post: "Sketch the graph to which the Fourier series converges."
The original problem is this:
for all x
Well you just have to sketch the curve given. Sketch the curve between -1 and 1, then as it is periodic with period 2 just tag copies of the sketch between -1 and 1 on at each end.
It is a saw tooth waveform going from at to at and from at to at , then repeats ...
Note there appears to be a mistake somewhere, that series does not correspond to that function.
CB
I know how to sketch the curve of the initial set of equations. What I don't know is how to sketch the graph to which the resulting Fourier series converges.
I went over my work again, and I can't find any mistake. I'll show the major steps:
Your limits of integration are wrong you should have sums of an integral from -1 to 0 and one from 0 to 1.
If you have done the working correct it will converge to the original function. If you are going to plot the partial sums you will need some computational system to do the calculations for any thing other than two or three terms.
CB
Sonofa... <smack>
Thank you for pointing that out! I completely missed that part of the technique when it was discussed.
Okay, so now that I've made that correction and redone everything, I now get this:
Looking at that, doesn't that just converge to zero?
You can get WolframAlpha to plot the partial sums of this without difficulty:
sum for n=1 to 10 ( -2/(n*pi) sin(n*pi*x) ) - Wolfram|Alpha
CB