Can anyone explain to me how to use the voltage division to construct the transfer equation? I'm baffled, it gives me the answer but I don't understand why. Any help is appreciated, thank you!

 

Step 1.
Replace all elements with their equivalent impedances.
So what do you have?
You have Ri, sL, 1/(sC) (there is an error here, the s is missing in your book), R.

Step 2.
Let us form a voltage divider.
The simplest voltage divider is made of two resistors.
Since all the elements are in series, we can simplify the circuit to have two "resistors". We will call them R and R1. R is the resistor where we take the Vo(s).
R1=Ri+sL+1/(sC)

Step 3.
Setup the voltage divider equation.
Vo(s)=[R/(R+R1)]*Vi(s)

Step 4.
Transfer function.
H(s)=Vo(s)/Vi(s)=R/(R+R1)
H(s)=R/(R+R1)

The rest is history.

 

Very good, thanks a ton, I see where I was getting confused. The highest exponent in the denominator has to have a coefficient of 1. I was looking at a standard RC circuit and tried to transition it over and couldn't see where I was screwing up. I really appreciate all your help!