I like this!
Immediately I can also see that 1/5 + 1/25 + 1/125 + ... = 1/4
To generalize: 1/x + 1/(xx) + 1/(xx*x) + ... = 1(x+1)
'Proved' by looking at a picture :-)
Immediately I can also see that 1/5 + 1/25 + 1/125 + ... = 1/4
To generalize: 1/x + 1/(xx) + 1/(xx*x) + ... = 1(x+1)
'Proved' by looking at a picture :-)
Can someone please post a non-visual proof of why this is the case? In the meantime, I am working on figuring out my own.
1/4 is 1/3 of 3/4
1/4 of 1/4 is 1/3 of 3/4 of 1/4
etc.
1/4 = 1/3*3/4
1/4^2 = 1/3*3/4*1/4
1/4^3 = 1/3*3/4*1/4^2
etc.
(1/4^1 + 1/4^2 + ...) = 1/3 * 3/4 * (1 + 1/4^1 + 1/4^2 + ...)
<=>
(1/4^1 + 1/4^2 + ...) = 1/3 * 3/4 + 1/3 * 3/4 * (1/4^1 + 1/4^2 + ...)
<=>
x = 1/3 * 3/4 + 1/3 * 3/4 * x
<=>
x - 1/4 x = 1/4
<=>
3/4 x = 1/4
<=>
x = 4/3 * 1/4
<=>
x = 1/3Yes, I did indeed mean 1/(x-1). Thanks for the correction.
S(n) = 1/n + 1/n^2 + ...
= 1/n ( 1 + 1/n + 1/n^2 + ...) <--needs more justification in a rigorous proof
S(n) = 1/n ( 1 + S(n) )
Simple algebra from here:
n * S(n) - S(n) = 1
S(n) = 1 / (n-1)