2

I saw a proof in a comment on a previous MSE question yesterday and I can't stop thinking about it. It looked like the following:

$0 = (1-1) + (1-1) + \cdots = 1 + (-1+1) + (-1 + 1) + \cdots = 1 + 0 + 0 + \cdots = 1$

Where does the proof go wrong? Obviously, 0 does not equal 1, so there must be some error here.

curiousmathstudent
  • 41
  • 2
    The series $1-1+1-1+\cdots$ does not have a well-defined value (by the usual definition). However, using Cesaro summation, its value is $1/2$. – angryavian May 20 '16 at 04:18
  • The problem is that though $(1 - 1) + (1 - 1) + \cdots $ is a meaningful expression, $1 - 1 + 1 + \cdots$ is a divergent series, which does not actually refer to a real number. – AJY May 20 '16 at 04:19
  • Interesting! I'm only a high school student so I don't really know what a divergent series is, or what Cesaro summation is, but I'll certainly remember this question when I do learn those terms in the future. I'm curious about the down vote though. What rules did I break, for future reference? – curiousmathstudent May 20 '16 at 04:21

3 Answers3

3

Note that $$1-1+1-1+1-1+\cdots$$ doesn't converge, so the associative law doesn't work.

Another example: $$S=1+2+4+8+\cdots=1+2(1+2+4+\cdots)=1+2S,$$ which implies $S=-1$. This clearly is wrong, because this serie is disvergent, and $S$ is not well defined.

Paul
  • 20,553
1

Consider the sequence $s_1=1, s_2=1-1=0, s_3=1-1+1=1, s_4=1-1+1_1=0,$ etc. Which does not converge,but alternates between $0$ and $1.$ what your example shows is that if a sequence $s_1=a_1, s_2=a_1+a_2,s_3=a_1+a_2+a_3,$ etc, does not converge, then we cannot insert brackets at will in the series $a_1+a_2+a_3+...$ and get consistently equal results. At the heart of this is the exact definition of an infinite sum (a limit) which is not at all the same thing as a finite sum.

DanielWainfleet
  • 57,985
0

The error lies in changing the order of the infinite summation. You saw that clearly $$(1-1)+(1-1)+(1-1)+ \dots = 0,$$ but in the next step it equalled one. This implies that there is something inherently flawed in the associative property for infinite sums.

Drew Christensen
  • 500