>I am looking for a proof that:
>1/2 + 1/4 + 1/8 + 1/16 + 1/32 + .....
>and so on for infinity ..... =1
The unknown number is always called x :
x=1/2 + 1/4 + 1/8 + ...
First put the first term outside parentheses:
x=1/2 + (1/4 + 1/8 + 1/16 + ...)
Next move the factor (1/2) outside the parenthesis:
x=1/2 + (1/2)(1/2 + 1/4 + 1/8 + ...)
Then observe that the contents of the second pair of
parentheses exactly equals your original series:
x=1/2 + (1/2)x
Now multiply both sides of this equation by 2:
2x=1 + x
Finally subtract x from both sides:
x=1
>There you are!
>
>>I have some friends who are very, VERY difficult to
>>convince.
>My friends are also hard to convince ! (see the other thread
>on this subject)

Infinity is a difficult concept - it would never exactly equal one after any finite period of time but, in the limit, at infinity, it would exactly equal 1.

> meaning that it would be
>.99999999999..., so there's an infinitely small difference
>between that answer and one. An infinitely small difference
>is not the same thing as no difference, I think.

I think it is. Using the same argument as before,

let           X = 0.9999...
so          10X = 9.9999...
so 9X = 10X - X = 9.0000... = 9

When you say 1/infinity, what that means is
"the limit, as n tends to infinity, of 1 / n"
which is, in fact, zero.

Rich

Does that make any sense? I also think that you can show infinity by 1/.0000... (repeating overscore thing)...1, though I'm not sure that's legal.

What you are discussing is the idea that in the limit process e.g. for any constant C the limit of C/n as n tends to 0 is infinity.

Here's the difficulty with that approach. Consider all the points on that straight line. They have length 0. But the union of all the points on the line is 1. So if the length of the line can always be broken up into the sum of its lengths, I can add 0 to itself an infinite number of times to get 1! Not good. This is the heart of Zeno's paradox: this argument works for 1/2 + 1/4 + ... = 1, but not for sum over all points of 0 = 1.

You are implicitly using what is known as "countable additivity" in your argument, the proof of which relies on the Induction Axiom. While most mathematicians use this axiom with impunity, it can never be proven empirically.

Mark Huber