



 
Andy, First, it's important to understand that the "sum" of a series such as 1+ 1/2 + 1/4 + ... or even 1 + 0 + 0 + ... cannot be derived directly from ordinary addition; it is necessary to introduce a rule that assigns sums to these.
It may also be shown that no rule obeying these can give a finite value to all series: for instance $s[1+1+1+...] = 1 + s[1+1+1+...]$ which is impossible. The standard rule for summing series does not assign a value to 1+2+4+... , because the sequence of partial sums (1,3,7,...) does not approach a limit. However, it can be shown that it  and any other rule with the properties above  can be extended consistently to a summation rule that does sum this series, and that the extended rule must always take the series to the value 1. Informally this is because $(12) S(1+2+4+...) = S[(1+2+4+...) (0+1+2+4+...)] It can be shown that if you start only with a summation rule like this and Hardy's axioms you cannot reach a contradictory conclusion: details are in my paper "Formal Summation of Divergent Series" (R. Dawson), Journal of Mathematical Analysis and its Applications 225(1998) 532541. However, this is NOT what is usually meant by the sum of a series, and standard summation by limits (especially of absolutely convergent series) has good properties that the extended method loses. Good Hunting!
 


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