A close approximation for square root of 2.

Boorglar

By chance I stumbled on this "almost" equality:

[itex]\frac{1}{5}(1/2+2/3+3/4+4/5+5/6+6/7+7/8+8/9+9/10) ≈ √2 - 7.2×10^{-6}[/itex]

I'm just wondering, are these funny coincidences simply, well, coincidences or is there some kind of explanation?
I've see a ton of other funny stuff like [itex]e^{\pi} - {\pi}≈19.99909998[/itex].
How likely is it for a relatively simple expression involving unrelated constants to work out almost nicely? Is it actually easy to come up with these meaningless things?

moonman239

I don't think it's very likely. Now, if someone could by chance stumble upon a way to calculate pi to any decimal place (i,e, whoever runs the equation can calculate pi to as many decimal places as he pleases) or prove us all wrong by showing that pi can be expressed as a ratio, that would be great.

SteveL27

There are many known series for pi.

http://mathworld.wolfram.com/PiFormulas.html

As far as the OP's question, note that the series is divergent. It just happens to be the case that a particular partial sum is close to something else. Take one more term of the series and the coincidence disappears. It seems that would happen fairly often ... it's not as big a coincidence as some series or expression actually converging close to something else, like the ones here ...

http://en.wikipedia.org/wiki/Mathematical_coincidence

Cardinal

nice I like that with [itex]e[/itex]^{[itex]\pi[/itex]}-[itex]\pi[/itex]

I accidentally found [itex]\sqrt{79}=8.888194417...[/itex] which I like to say/present as [itex]\sqrt{69+10}[/itex] but I don't think these things are coincidental , probably we have a lot more to learn.

e^(i Pi)+1=0

∏=[itex]\frac{c}{d}[/itex]