Simplifying the Fourier Series Function: Tips & Tricks

[darkmagic]

Homework Statement

f(x) = 1 0<t<1
= -1 1<t<2

How can I simplify this given that function(on the attachment).

What I mean is that how can I write the function in any other way?

In addition, How can I know if the function can be written in other form?
How can I write the function in other form?

Homework Equations

The Attempt at a Solution

 

[HallsofIvy]

If n is even, 1- (-1)ⁿ= 1- 1= 0! If n is odd, 1- (-1)ⁿ= 1- (-1)= 2.

So
$$\frac{2}{\pi}\sum_{n=1}^\infty \frac{[1- (-1)^n] sin(n\pi t)}{n}$$
is just
$$\frac{2}{\pi}\sum \frac{2 sin(n\pi t)}{n}$$
where now the sum runs only over odd n. One way to show that is to use 2n+1 rather than n in the body of the sum. That way, as n goes over all non-negative integers, 2n+1 goes over all positive odd integers:
$$\frac{4}{\pi}\sum_{n=0}^\infty \frac{sin((2n+1)\pi t)}{2n+1}$$

 

[darkmagic]

Can 2n+1 be 2n-1 provided that n=1 to infinity?
How can I know if the function can be converted in some form?