Given the series Σ⁰n=0n2n, you need to show that it converges to 2. You were given a hint to take the derivative of Σ⁰n=0xn and multiply by x, which gives Σ⁰n=1nxn, or Σ⁰n=0nxn. Clearly, if you take x=1/2, the series is Σ⁰n=0n2n. How do you proceed from here?
