Infinite number of mathematicians enter a bar.
The barman asks:
-Hi all, what can I bring for you?
The first mathematician orders a beer,
the second says: - Half beer for me.
The third mathematician says: - Half of what the previous guy ordered.
The fourth: - Half of the previous
The fifth: -Half of the previous, too.
and so on...
Barman (really upset): -Stop it, idiots
and serves them two beers!!
Background
The joke exploits the notion of convergence. For example, imagine that we sum $1+2+3+...$ till infinity. (In mathematical compact notation it is written as $\sum_{n=1}^{\infty} n$).
Obviously this sum becomes bigger and bigger and gets infinity; in mathematics we say that "the sum diverges".
However, say we sum $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...$ till infinity, in other words we add a term which is the half of the previous one till infinity (what the mathematicians did ordering beers). In compact notation we would write $\sum_{n=0}^{\infty}(\frac{1}{2} )^n$
Now we observe (with the assistance of a calculator) that this sum does not grow incredibly fast, but rather approaches a number. This is the number 2, and every time you add a new "half of the previous" you get even closer to the value of two. In scientific jargon, this notion of approaching a finite number is called, convergence, and the sum is convergent.
Is the number half a special number? Do other series converge to finite numbers? The answer is yes and the relative concept is the convergence of the geometric series. For any number $r$ between $-1$ and $1$, the series
$\sum_{n=0}^{\infty} r^n = \frac{1}{1-r}$,
which is called the geometric series, is always convergent to the number $\frac{1}{1-r}$. To convince yourself that the formula works try the case where $r=1/2$ and you will find the value of $2$.
Did you like the notion of convergence and geometric series? For more information check out:
http://en.wikipedia.org/wiki/Geometric_series
http://mathworld.wolfram.com/GeometricSeries.html
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