4. Let L E R. A series 2::%"=0 ak is said to be Abel summable to L if and only if 00 lim '" akrk = L.

r---+l-~

k=O

(a) Let Sk = 2::7=0 ak· Prove that 00 00 00 L akrk = (1- r) L Skrk = (1- r)2 L(k + 1)a'krk , k=O k=O k=O provided that anyone of these series converges for all 0 < r < 1.

(b) Prove that if 2::%"=0 ak is Cesaro summable to L, then it is Abel summable to L.

(c) Prove that if f is continuous, periodic, and of bounded variation on R, then Sf is Abel summable to f uniformly on R.

(d) Show that if ak ;:::: 0 and 2::%"=0 ak is Abel summable to L, then 2::%"=0 ak converges to L.