Infinite series simplification

Question

Is there a clean way to simplify the following series:

$$ 0.5^2(1-2*0.5^2)+0.5^3(1-2*0.5^3)+\cdots +0.5^k(1-2*0.5^k) $$ where k = 1, 2, ... ∞

Using R led to the convergence point = 1/3

n = 10000
x = rep(0,n-1)
for (i in 2:n){
  print(i)
  x[i]= (0.5^as.numeric(i))*(1-0.5^as.numeric(i)-0.5^as.numeric(i))
}
sum(x)

Answer 1

$$ \sum_{k=2}^\infty \left\{\frac1{2^k}-2\frac1{2^{2k}}\right\}=\frac14\sum_{k=0}^\infty\frac1{2^k}-\frac24\sum_{k=0}^\infty\frac1{4^k}=\frac14\frac1{1-\frac12}-\frac12\frac1{1-\frac14}=\frac12-\frac16=\frac13 $$