Find $\mathcal{L}(F(t))$ where $F(t)$ is the perioidic function shown graphically below:
I know a few basic things, but don't know where to get started. For example, I know that the period, T = 2, and that $F(t)$ fluctuates from -1 to 1. Other than that, not too sure where to start.
Note: this is question 87 from Spiegels Laplace transform textbook.
$\mathcal L(F(t) (x)=\int_0^{\infty} e^{-tx} F(t)\, dt$. Split the integral into integrals over the intervals $(0,1),(1,2)$ etc. You get $\frac {1-e^{-x}} x-\frac {e^{-x}-e^{-2x}} x+\frac {e^{-2x}-e^{-3x}} x-\cdots$. You can express this in simple form by adding two geometric series. I will leave that part to you.