A nice proof without words

1Aug/114

Here's a very cool proof without words by Burkard and Marty.

Can you figure out what this proof without words illustrates?

.mau.
August 1st, 2011 - 07:03

this is basically the result found by a Bernoulli (I don’t remember which one), right?
Kelly John Rose
August 1st, 2011 - 08:09

That \sum_{x=1}^{\infty} 1/x^k converges for any k >= 2 and is < 1

It doesn't provide what it converges to though.

What's interesting is I wonder if this idea could be extended to show that harmonic series don't converge.
Matt Henderson
August 1st, 2011 - 15:52

In a way, it’s cheating a little in that it sort of assumes 1/2 + 1/4 + 1/8 + … = 1, so all the columns actually do fit into the square. But I think you can kind of see they do fit in the square.
I guess you could deduce from it (Pi^2)/6-1<1 so Pi<3.47
Naeemul Hassan
August 2nd, 2011 - 13:01

(1/2^2)+(1/3^2)+(1/4^2)+(1/5^2)+(1/6^2)+…… < 1

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