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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 pi
by jakkals 17y ago
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 :-)
- zackattack 17y agoYou mean, = 1/(x-1) ;) 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.
- swapspace 17y agoInfinite GP: a/(1-r)
- zackattack 17y agoAh, true. http://en.wikipedia.org/wiki/Geometric_series#Formula http://en.wikipedia.org/wiki/Geometric_series#Formula
- I_got_fifty 17y agoThis wouldn't happen on Digg.
- jonsen 17y agoYou can get the proof idea from the picture: 1/4 is 1/3 of 3/4 1/4 of 1/4 is 1/3 of 3/4 of 1/4 etc. In math: 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. Summing equations: (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 = 1/3
- jakkals 17y agoRats! Yes, I did indeed mean 1/(x-1). Thanks for the correction.
- ewjordan 17y agoNot a rigorous proof (I'll leave that to further investigation): 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)
- deleted 17y ago[deleted]