Question: A really good friend of mine is an elementary school math teacher. He is turning 50, and we want to put a mathematical expression that equals 50 on his birthday cake but goes beyond the typical "order of operations" problems. Some simple examples are 
$$e^{\ln{50}}$$
$$100\sin{\frac{\pi}{6}}$$
$$25\sum\_{k=0}^\infty \frac{1}{2^k}$$
$$\frac{300}{\pi^2}\sum\_{k\in \mathbb{N}}\frac{1}{k^2}$$

What are some other creative ways I can top his cake? 

I should note that he is an elementary school teacher. Now he LOVES math, and I can certainly show him a lot of expressions. I don't want them so difficult that it takes a masters degree to solve, but they should certainly be interesting enough to cause him to be wowed. Elementary functions are good, summations are also good, integrals can be explained, so this is the type of expression I'm looking for...

EDIT:: I would make a note that we are talking about a cake here, so use your judgement from here on out. Think of a normal rectangular cake and how big it is. Hence, long strings of numbers, complex integrals, and long summations are not going to work. I appreciate the answers but I need more compact expressions.

Answer: 
Whenever one of my friends has a birthday, I find out how old they get and then I visit [their number on Wikipedia](http://en.wikipedia.org/wiki/50_%28number%29)

For your friend, I would write something like this:

> 
> 50 is the smallest number that is the sum of two non-zero square numbers in two distinct ways: $1^2 + 7^2$ and $5^2 + 5^2$. It is also the sum of three squares, $3^2 + 4^2 + 5^2$. It is also a Harshad number and a nontotient and a noncototient. 50 is the aliquot sum of 40 and 94. 50 is also the atomic number of tin and fifth magic number in nuclear physics.
> 
> 
> 

While many of the things on Wikipedia are not Mathematical expressions, and some of it is way too long to write on a cake, I am certain that this will brighten his day if you tell him this stuff!

As for a mathematical expression, I'd go with either $3^2 + 4^2 + 5^2$ because I find it simple but elegant, or fill the cake with stuff like "Harshad number, 5th magic number in nuclear physics, nontotient", etc. and see if he can figure out how old he is.
Some solutions:
$$\dfrac{50(-1)^{-i}}{i^{i^{2}}}=50$$
$$50=\sum\_{n=1}^{10}(2n-1) \mod 10$$
$$50=1212\_3$$
$$\left(4+\frac{4}{4}\right)\dfrac{\binom{4}{4-\frac{4}{4}}}{0.4}=50$$