How do you derive the expansion for k/3^k

(1/3) + (2/3^2) + (3/3^3) + (4/3^4) +... + (k/3^k)+ ...

Use the geometric series
1 + x + x^2 + x^3 + ... + x^k + ... = 1/(1 - x).

Differentiate both sides:
1 + 2x + 3x^2 + ... + kx^(k-1) + ... = 1/(1 - x)^2.

Multiply both sides by x:
x + 2x^2 + 3x^3 + ... + kx^k + ... = x/(1 - x)^2.

Let x = 1/3 (which is valid, because |x| = 1/3 < 1):
1/3 + 2/3^2 + 3/3^3 + ... + k/3^k + ... = (1/3) / (1 - 1/3)^2 = 3/4.

I hope this helps!