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thorx440

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## Homework Statement

You have series expansions of the function f(x) = 0 from 0 to .5, and 1 from .5 to 1 : the halfrange cosine series, the half-range sine series, and the Fourier series. For each of these series, find the actual sum of the series at x = 0, and x =1/2, and x =1

## Homework Equations

1/2 [f(x+) + f(x-)] if discontinuous

f(x) if continuous

## The Attempt at a Solution

I found the Fourier half range sine, and cosine expansions. But I was unsure of how to continue. Looking at the graph, I can see that the function is continuous from 0 to 1, then repeats, being discontinuous 0,1,2,3...etc.

For x = 0, it is discontinuous, so I used the first equation and found that f(x+) = 0 and f(x-) = 1, so the actual sum of the series at x = 0 is 1/2.

for x = 1/2, the point is continuous on the function, so the actual sum of the series is equal to f(1/2). Plugging f(1/2) into the Fourier series expansion that I found leaves me unable to solve it since I still have an infinite series.

The Fourier expansion is equal to

1/4 + summation from 1 to infinity of [(-1)^(n+1)/(n*pi) + cos(n*pi/2)/(n*pi)] cos(n*pi*x) + [- sin(n*pi/2)/(n*pi)]sin(n*pi*x)

(I apologize for the mess of equation, I have no idea how to format it so it looks nicer)

edit: looking at it again makes me start to think I did the first part wrong...unless the actual sum of the series at x=1 and x = 0 are the same and both equal 1/2

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