Chapter 2, Section 2.6

We want to derive a series expansion for Cos[g]=((1–r/a)+(e^2))/(e(1–r/a)+e and show that it is equal to Cos[M] to order e.  First we define Cos[g] in terms of mra=1–r/a and e (see Eq.(2.103)).

cosg  =  (mra + e^2)/(e*mra + e)

Now derive a temporary series expansion for r/a using Eq.(2.81).

troa = 
Normal[Expand[1+e^2/2-2e*
Series[Sum[(1/s^2)*
D[BesselJ[s, s e],e]Cos[s*M],{s,1,6}],
       {e,0,6}]]]

Use this series to derive 1–r/a taking terms up to fourth order in e.

mra = 
1-Collect[Expand[Series[Normal[troa],{e,0,4}]], e]

Now look at the expression for Cos[g] ...

cosg

... and expand this as a series in e.  Note that there are no first order terms and so g=M to order e.

Expand[TrigReduce[Normal[Series[cosg,  {e, 0, 2}]]]]