Sums of converging of sequences

Question

For what values of the variable does the series converge? Use the properties of geometric series to find the sum of the series when it converges. 12x+24x^2+48x^3+96x^4+⋯

Answer 1

This is geometric infinite series with a₁ = 12x and common ratioo r = 2x. It converges when - 1 < r < 1 ,

1. 1 < 2x < 1 or - 1/2 < x < 1/2. The sum opf this series S = a₁/(1 - r) = 12x/(1 - 2x) for - 1/2< x < 1/2

Answer 2

-1 < x < 1, x can't be greater than one, or less than minus one

factor out a 6 to get

6(2x + 4x^2 + 8x^3 ....)

then

sum = 6 times first term times 1/(1-2x) = 12x/(1-2x)

each term is 2x times the previous term. 2x = r in the general formula sum = first term/(1-r)