I am not sure I am doing this exercise good
1) $p_n $ | all parts are pairs different
and
2) $p_n $| all parts are not higher than $m$
I found these functions in book, first is:
$$ \prod_{i=1}^\infty (1+x^i)$$
And second:
$$ \prod_{i=1}^\infty (1-x^{2i-1})^{-1}$$
And I have to proof that those two functions are equals.
so:
Then I found something like this:
$$ (1+x^i) = \left(\frac{1-x^{2i}}{1-x^i}\right) $$
Then i calculated right side to check it
$$ \left(\frac{1-x^2}{1-x^1}\right)\left(\frac{1-x^4}{1-x^2}\right)\left(\frac{1-x^6}{1-x^3}\right)\left(\frac{1-x^8}{1-x^4}\right)\cdots = \frac{1}{1-x^{2i-1}} $$
So this seems to be proof for this example but i dont know how i can find it by myself that
$$ 1+x^i = \frac{1-x^{2i}}{1-x^i} $$
I dont know how to get this step. I will be very thankful for every help.