<s>[INST] Euler's partition method. How does someone use it? I came across partitions recently and am not very much informed about it but I have a question regarding Euler's method for this. I came to know about this formula from a YouTube video so, it may not be the full equation.
Symbols:
P(n) -> Partition of n
$\pi$ -> Product
n -> Any number
$\sum_{}^\infty (P(n)*x^n) = \prod\limits_{k=1}^\infty (\frac{1}{1-x^k})$
I can't understand:

*

*What does 'x' mean in this expression

*How can someone calculate partition with this. 
The video I learnt this from is this.

Note: I am not informed regarding this subject so, I may have the question all wrong. I only want to know how this works. Today, I don't think this formula is used today after Ramanujan's work. I just want to know how the people before used it.
 [/INST] This example supplements MAAvL's answer and is an expansion of dust05's comment.  The following is a bit pedantic, with including exponents 1 and writing $(1+x^2+x^{2+2}+\cdots)$ rather than $(1+x^2+x^4+\cdots)$, for example, but the idea is to make the connection to partitions very concrete.
\begin{align*}
&\sum_{n \ge 0} P(n) x^n = \prod_{k \ge 1} \frac{1}{1-x^k} \\
& = \left(\frac{1}{1-x}\right) \left(\frac{1}{1-x^2}\right) \left(\frac{1}{1-x^3}\right) \left(\frac{1}{1-x^4}\right) \cdots \\
& = (1+x^1+x^{1+1}+\cdots)(1+x^2+x^{2+2}+\cdots)(1+x^3+x^{3+3}+\cdots)(1+x^4+x^{4+4}+\cdots)\cdots \\
& = 1 + x^1 + (x^2 + x^{1+1}) + (x^3 + x^2 x^1 + x^{1+1+1}) + (x^4 + x^3 x^1 + x^{2+2} + x^2 x^{1+1} + x^{1+1+1+1})\cdots\\
& = 1 + x^1 + (x^2 + x^{1+1}) + (x^3 + x^{2+1} + x^{1+1+1}) + (x^4 + x^{3+1} + x^{2+2} + x^{2+1+1} + x^{1+1+1+1})\cdots\\
& = 1 + x + 2x^2 + 3x^3 + 5x^4 + \cdots
\end{align*}
Also, this generating function for $P(n)$ is used all the time in contemporary research on integer partitions (where it is often written using the $q$-Pochhammer symbol as $1/(q;q)_\infty$ or just $1/(q)_\infty$).  As per the comments and links, though, there are other ways to compute $P(n)$ exactly (Euler's recurrence using the pentagonal number theorem) and asymptotically (the Hardy-Ramanujan-Rademacher formula).
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