Find the power series expansion of the following in powers of x and state what values of x it is valid for

Find the power series expansion of the following in powers of x and state what values of x it is valid for

a) ln(3+2x)

b) x^2/(a-x^2)

Are there 2 methods for this? A friend gave me her notebook from when she took the class and she had a method that did not use integration and the teacher showed us examples in class with integration. What I tried starting for a was:

ln(3x+2) = ln(2((3/2)x+1)) = ln(2) + ln((3/2)x+1)
Let u = 3/2x
= ln (2) + ln(1+u)
= ln(2) + sum(1 to inf) (-1)^(k+1) * u^k/k
= ln(2) + sum (-1)^(k+1) (3/2x)^k/k
......
but I have no idea if that is correct "so far" and what to do after that point. My teacher and my friend skipped a lot of steps so I am not sure how to follow the process. Thanks for your help!!! :)

a) This looks fine; I'd write it as
ln(2) + Σ(k=1 to ∞) (-1)^(k+1) (3x/2)^k/k.

This is valid for -1 ≤ 3x/2 < 1
==> -2/3 ≤ x < 2/3.

(By the way, one can obtain the log series from integrating the geometric series.)

b) Use the geometric series 1/(1 - t) = Σ(n = 0 to ∞) t^n, convergent for |t| < 1.

Let t = x^2/a:
1/(1 - x^2/a) = Σ(n = 0 to ∞) (x^2/a)^n, convergent for |x^2/a| < 1.
==> a/(a - x^2) = Σ(n = 0 to ∞) x^(2n)/a^n, convergent for |x| < √|a|.

Multiply both sides by x^2/a:
x^2/(a - x^2) = Σ(n = 0 to ∞) x^(2n+2)/a^(n+1), convergent for |x| < √|a|.

I hope this helps!