I was looking to analytically approximate the integral of a common bump function in topology:
$$ e^{\frac{1}{1-x^{2}}} $$
This problem can be quickly realized into being related to the distinct partition function from number theory by this formula, $q(n)$ being the distinct partition function:
$$ \sum_{n=0}^{\infty} q(n)x^{n} = \prod_{n=1}^{\infty} (1+x^{n}) $$
However, the product function that I need to consider is of the form:
$$ \prod_{n=1}^{\infty} (1+ax^{n})$$ With any $ a \in [0,1]$. So my question is whether there are any manipulations of the product such that the generating function can be used in some regard, my ultimate goal would be to entirely express it as a summation. Thank you so much I thought this problem was an interesting and easily comprehensible union of number theory and topology, and your help would be very appreciated.