Finding Fourier Expansion of f(t): Part b & c

[zyphriss2]

Find the Fourier expansion of one period of f(t)=1+t absolute value of t<1

I found this to be 1+2/pi Sigma(0 to infinity) ((-1^(n+1))/n)sinnpit by just the standard methods of the a0 an and bn formuals, which I know is correct

Now the parts I am having problems with is part b and c which our teacher has not covered much at all and I cannot find any help online.

Part b. Use the Fourier expansion of f to find the sum of the Series
Sigma(0toinfinity)(-1^n)/(2n+1)

Part C. If f denotes the function defined on (-infinity to infinity) by the Fourier series of f, find F(1)+F(-5)-3F(0)

 

[xepma]

Just to make things a bit more clear on the notation side, I take it your expansion is:

1+\frac{2}{\pi}\sum_{n=0}^{\infty}\frac{(-1)^{n+1}}{n} \sin{n\pi t}

As for question b:

Notice that if you set t equal to t_0 \equiv \frac{1}{2}, then

\sin{n\pi t_0} = \sin{\frac{n\pi}{2}}

which is zero whenever n is even and (-1)^{(n-1)/2 whenever n is odd... Do you see where I'm going with this?

part C:
I take it by F(0) you just mean f(0)? Have you tried filling in these numbers into the Fourier series?