Help with a series expansion in Marion & Thornton

[snatchingthepi]

Homework Statement

Finding the range of an object shot from a canon with air resistance of the form -k*m*v

Relevant Equations

Series expansion

So I'm on page 67 of Marion/Thornton's "Classical Dynamics of Particles and Systems" and I'm in need of some help. I understand that so far there's is an equation that cannot be solved analytically (regarding motion due to the air resistance and finding the range of the an object shot from a canon). So we got to take a series expansion of the exponential to help with this.

The problem is that I cannot for the life of me see how to arrange the terms to get the forms the author uses. Specifically I'm mystified about how 2.45 is coaxed into 2.47 using 2.46, and how 2.47 is made into 2.49 using 2.48.

 

[vela]

Just do what was described in the text. It may help to group things together to simplify the algebra.
\begin{align*}
T &= \frac{kV+g}{gk}\left(kT - \frac 12 k^2 T^2 + \frac 16 k^3 T^3\right) \\
1 &= \left(\frac{kV}{g}+1\right)\left(1 - \frac 12 (kT) + \frac 16 (kT)^2\right)
\end{align*} Solve for the term linear in $T$.