A very cute algebraic way to see this is to do arithmetic in base 4. In base 4 the series on the left is:<p>0.1 + 0.01 + 0.001 + ... = 0.111... (recurring)<p>Multiplying the right hand side by 3 gives 0.3333... = 1, and so the original series must have just been 1/3.
I like this one better.
<a href="http://web.mat.bham.ac.uk/pgweb/random/2009/04/proof-without-words/" rel="nofollow">http://web.mat.bham.ac.uk/pgweb/random/2009/04/proof-without...</a>
I like this!<p>Immediately I can also see that 1/5 + 1/25 + 1/125 + ... = 1/4<p>To generalize:
1/x + 1/(x<i>x) + 1/(x</i>x*x) + ... = 1(x+1)<p>'Proved' by looking at a picture :-)
While pedantic, the above "expression" is NOT = 1/3<p>1/3 is the limit, as the sum of n=1 to n -> infinity, of (1/4)^n<p>The "result" converges towards 1/3.
You can get as close to 1/3 as you like, but the result will never quite equal 1/3.<p>Cheers
Dion.