SEARCH HOME
 Math Central Quandaries & Queries
 Question from Andy, a student: In this minutephysics video, it's claimed that 1+2+4+8....= -1 Is this true, and if so, how? http://www.youtube.com/watch?v=kIq5CZlg8Rg

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

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.