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.

Following G. H. Hardy, it is usual to require that any such rule should be well behaved under termwise addition of series,

$s[(a_0+b_0) + (a_1+b_1) + ...] = s[a_0+a_1+....]+ s[b_0+b_1...]$

scalar multiplication

$s[ca_0 + ca_1 + ca_2 + ...] = c s[a_0+a_1+a_2+...]$

and shift:

$s[0+a_0+a_1+a_2...] = s[a_0 + a_1 + a_2 ...]$

as well as reducing to finite summation - equivalently,

$s[a+0+0+0...] = a$

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

$(1-2) S(1+2+4+...) = S[(1+2+4+...) -(0+1+2+4+...)]
= S[1+0+0+... ] = 1.$
So for this rule S,
$S(1+2+4+...) = 1/(1-2) = -1.$

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) 532--541.

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!
RD