8. A series 2::'0 ak is said to be Cesaro summable to an L E R if and only if n-1 ( k) an := 2: 1 -:;;: ak k=O converges to L as n ~ 00.

(a) Let Sn = 2:~:~ ak. Prove that n

for each n EN.

(b) Prove that if ak E Rand 2::'0 ak = L converges, then 2::'0 ak is Cesaro summable to L.

(c) Prove that 2:%"=0 ( -l)k is Cesaro summable to 1/2; hence the converse of (b)

is false.

(d) [TAUBER]. Prove that if ak 2: 0 for kEN and 2:%"=0 ak is Cesaro summable to L, then 2:%"=0 ak = L.