sumation from k=0 to inf of (-1)^k * k! * (b+x)^[(-1)(k+1)] * (x-a)

valid from (0,1/(b+a)]

Sorry, that is not right ! From where do you have b+x as well as x-a ? What are the steps that you used to obtain the above step ?

You have \(\displaystyle 1/(b+x)\)

Step 1 :- What do you have after making the necessary substitutions as explained by my hint? Remember that you want it as powers of x-a.

Step 2 :- How do you expand \(\displaystyle 1/(1+y)\) ? In this problem,what do you have in place of y ?

Later we can discuss the validity of convergence.