Please go through the power series steps in detail as I am h

Please go through the power series steps in detail as I am having trouble getting this concept. Thank you so much for your help.

Solution

Start with the geometric series f(t)=1/(1 - t) = (k=0 to infinity) t^k. (This series converges for |t| < 1.)


Letting t = 12x:
1/(1 - 12x) =(k=0 to infinity) 12^k x^k.
(Now the series converges when |12x| < 1 ==> |x| < 1/12.)

Differentiate both sides:
12/(1 - 12x)² = (k=1 to infinity) k .12^k x^(k-1).
(The series converges when |x| < 1/12.)

Multiply both sides by x²/12:
x²/(1 - 12x)² = (k=1 to infinity) k 12^(k-1) x^(k+1).

let k+1=n,then

=(n=0 to infinity).(n-1).12^(n-2).x⁽ⁿ⁾
b)(The series still converges when |x| < 1/12. So, the radius of convergence is 1/12.)