Let L R A series k 0 ak is said to be Abel summable to L if and only if a) Let Sk j 0 ak Prove that provided any one of these series converges for all 0 b) Prove that if k 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 variatio...

a Fix N N and r 0 1 By Abels Transform If converges then S N r N 0 as N and we have verified the fir ...

Let L R. A series k=0 ak is said to be Abel summable to L if

Question:

Let L ˆˆ R. A series ˆ‘ˆžk=0 ak is said to be Abel summable to L if and only if

a) Let Sk = ˆ‘ˆžj=0 ak. Prove that

provided any one of these series converges for all 0 b) Prove that if ˆ‘ˆžk=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 ˆ‘ˆžk=0 ak is Abel summable to L, then ˆ‘ˆžk=0 ak converges to L.