FtG (02/01/83)
There have been several people who have questioned my statement that the series 1/2-1/2+1/2... sums to 1/4 as any calculus text will tell. I really should have said that 1-1+1-1... has a LIMIT of 1/2 as any GOOD calculus text will tell. For the doubters, see page 309 of "Advanced Calculus" by Widder, the section on Cesa`ro summability. This series is example A. The previous page (308) gives a good explanation of how not to manipulate infinte series. I think these sections conclusively back my earlier point that one should not confuse the limit of a series with the series itself. The operations one can perform on each are sometimes distinct. If one isn't careful, errors can arise. Or as Widder puts it "We MUST proceed by definition and not by analogy." Always ready to diverge... FtG