I would like to have some help with the next problem:

I'm trying to learn how to develop the actual function in the power series. After reading my book, I want to check if I understand correctly what is the process of developing the actual function given. So here's how I understand what I need to do to develop the actual given function $ f $:

1) Check if the given function $ f $ is infinitely different and where.

2) Choose the point $ x_0 $ in which we will develop the function.

3) Check if the given function $ f $ is continuous with all its derivatives, until the $ n $order 3, in some quarters of the point $ x_0 $. If this is accomplished, we have that we can write $ f (x) = P_n (x, x_0) + R_n (x) $.

4) Check if $ lim_ {n to infty} R_n (x) = $ 0.

5) Check if ## EQU1 ## $. This means that we have to check the convergence of the Taylor series that we have obtained and calculate the sum of the series if the series is convergent.

6) If all conditions are met, then we can say that this function $ f $ can be developed in the power series $ sum_ {n = 0} ^ { infty} frac {f ^ {(n)} (x_0)} {n!} (x – x_0) ^ n $ and we can call this analytic function.

Please, could you tell me if I understand this process correctly and if not, where am I wrong?