... so that...
It is evident that (2) is a 'telescopic series' so that is...
Let lim n-> ∞ a_n = L. Then, let f(x) = ∑ from 0 to ∞ of (a_n)(x^n). Show that the lim x-> 1 (1-x)f(x) = L.
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.