1. The problem statement, all variables and given/known data Let lim n-> ∞ a_n = L. Then, let f(x) = ∑ from n=0 to ∞ of (a_n)(x^n). Show that the lim x-> 1 (1-x)f(x) = L. 2. Relevant equations 3. The attempt at a solution This one is pretty far over my head. I know at some point you're supposed to use Abel/SBP, but here is what I have so far. Let |a_n| go to |L|. Then, using the ratio test, let |a_n|^(1/n) go to |L|^(1/∞) = |L|^(0) = 1. Then, from the Ratio test, we can see that the series will converge for |x| < 1. Take ∑ from 0 to ∞ of (a_n)(x^n). Then, multiply through. So, we obtain, (1-x)∑ from 0 to ∞ of (a_n)(x^n) = ∑ from 0 to ∞ of (x^n - x^(n+t)). Taking b_n to equal (x^n - x^(n+t)) we can get ∑ (a_n)(b_n).. This is where I get stuck. I'm not really where to take it from here.