A series k=0 ak is said to be Cesdro summable to an L R if and

A series ˆ‘ˆžk=0 ak is said to be Cesdro summable to an L ˆˆ R if and only if

converges to L as n †’ ˆž.
a) Let sn = ˆ‘n-1k=0 ak. Prove that

for each n ˆˆ N.
b) Prove that if ak ˆˆ R and ˆ‘ˆžk=0 ak = L converges, then ˆ‘ˆžk=0 ak is Cesaro summable to L.
c) Prove that ˆ‘ˆžk=0 (- 1)k is Cesaro summable to 1/2; hence the converse of b) is false.
d) [TAUBER'S THEOREM]. Prove that if ak > 0 for k ˆˆ N and ˆ‘ˆžk=0 ak is Cesaro summable to L, then ˆ‘ˆžk=0 ak = L

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N-1 On :- ak