(Cesaro summability) Although- (5) gives the usual definition of infinite series, it is no...

(Cesaro summability) Although- (5) gives the usual definition of infinite series, it is not the only possible one nor the only one used. For example, according to Cesaro summability, which is especially useful in the theory of Fourier series, one defines

that is, the limit of the arithmetic means of the partial sums. It can be shown that if a series converges tosaccording to “ordinary convergence” [equation (5)], then it will also converge to the same value in the Cesaro sense. Yet, there are series that diverge in the ordinary sense but that converge in the Cesaro sense. Show that for the geometric series (see Exercise),

Exercise

(Geometric series)

Show that

is anidentityfor allx≠1and any positive integern,by multiplying through by 1 −x(which is nonzero sincex≠1) and simplifying.

The identity(5.1)can be used to study the Taylor series known as the geometric series according to

(5.1), its partial sumsn(x)is

Show, from (5.2), that the sequencesn(x)converges, asn→∞, for|x| < 1, and diverges for |x| > 1.

Determine, by any means, the convergence or divergence of the geometric series for the points at the ends of the interval of convergence,x =±1. NOTE: The formula (5.2) is quite striking because it reducessn(x)to theclosed form(1−lxn)/(1−x),direct examination of which gives not only the interval of convergence but also the sum function1/(1−x). It is rare that one can reduce sn(x) to closed form.

for allx ≠1, and use that result to show that the Cesaro definition gives divergence for all |x| > 1 and for x =1, and convergence for|x| < 1, as does ordinary convergence, but that forx=−1it gives convergence to 1/2, whereas according to ordinary convergence the series diverges forx= −1.