Determining power series and interval of convergence from function

Question

How do you find power series and interval of convergence for f(x)=3/(4+x) centered around x=0?

Answer 1

The sum of the geometric series ∑_{(from n = 0 to infinity)} arⁿ is a / (1-r), if -1 < r < 1, where a is the first term and r is the common ratio.

So, if we rewrite 3 / (4+x) as 3 / (4(1 + x/4)) = (3/4) / (1 - (-x/4)), we see that this is the sum of the geometric series with a = 3/4 and r = -x/4. We get convergence as long as -1 < -x/4 < 1.

So -4 < x < 4.

The power series is ∑_{(n=0 to infinity)} (3/4)(-x/4)ⁿ = 3∑_{(n=0 to infinity)} (-1)ⁿxⁿ / 4ⁿ⁺¹.