Wild About Math!Making Math fun and accessible

1Aug/114

A nice proof without words

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

Can you figure out what this proof without words illustrates?

1. this is basically the result found by a Bernoulli (I don’t remember which one), right?

2. 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.

3. 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

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