If the sum to infinity of the series: x + (2x/3) + (4x/9) + ... is fifteen, what is x?

We notice that we could re-write the
sum:

x + x*(2/3) + x*(2/3)^2 + ... + x*(2/3)^n, n->
infinite

We'll factorize by
x:

x[1 + 2/3 + (2/3)^2 + ...  +(2/3)^n] =
15

x*lim[1 + 2/3 + (2/3)^2 + ...  +(2/3)^n] =
15

x*lim [1-(2/3)^(n+1)]/(1-2/3) =
15

x*lim 3*[1-(2/3)^(n+1)] =
15

3x*lim[1-(2/3)^(n+1)] = 3x - lim (2/3)^(n+1) =
15

3x - 0 = 15

3x =
15

x = 5

The
value of x, for the sum of the infinite number of terms of the series is 15, is x =
5.