dap1 (02/04/83)
#R:rocheste:-57800:ihlpb:6200010: 0:129 ihlpb!dap1 Feb 3 11:41:00 1983 Cesaro summability is one thing and series convergence is another as any GOOD calculus text (including the one cited) will show.
berry (02/04/83)
#R:rocheste:-57800:zinfandel:7700002:000:1052 zinfandel!berry Feb 2 11:08:00 1983 I have not yet been able to read up on 'Cesaro summability', but I hotly (FLAME ON?) contend that the sequence 1/2 - 1/2 + 1/2 - 1/2 ... + 1/2*(-1)**(n-1) + ... has a limit IN THE GENERALLY ACCEPTED SENSE. let s-sub-n denote the sum of the first n terms, and then s-sub-2n is 0, and s-sub-(2n-1) is 1/2, for any n. then lim s n -> oo n does not exist, when 'lim' is as defined in chapter 1 of all 1st year calculus books (well maybe chapter 2), you know, the epsilon-delta stuff someone has summarized well. My reference is "Higher Mathematics for Engineers and Physicists" by Sokolnikoff and Sokolnikoff, McGraw Hill, 1941, page 5. They examine the series 1 - 1 + 1 - 1 ... which is just like the one under discussion, multiplied by 2. I'm gonna look up that Cesaro reference, you betcha, and I will report what I find. If I'm wrong, I'll admit it, but I bet given the Math 1 definition of limits, that sequence doesn't converge, nohow! Berry Kercheval Zehntel Inc. (decvax!sytek!zehntel!zinfandel!berry) (415)932-6900