# Difference of Divergent Products Converges?

I was thinking about series and was wondering how infinite products are dealt with. For example, consider this difference of two divergent products:

$$\prod_{n=2}^\infty \ln(n) - \prod_{n=2}^\infty (\ln(n)-1/n)$$

I don't know how to determine bounds for the difference, or prove if it converges to a limit. I remember one can take the logarithm of an infinite product to convert it into an infinite sum, but I don't think that strategy applies here because it is a difference of infinite products, not a single infinite product. I really don't know how to deal with the difference of products like this. If someone is familiar with this type of situation, please let me know.

I assume you meant to write $$~\displaystyle\lim_{N\to\infty}~\prod_{n=2}^N\ln(n)~-~\prod_{n=2}^N\bigg[\ln(n)-\dfrac1n\bigg],~$$ otherwise the expression

would make little sense, since $$\infty-\infty$$ is undetermined. Now, factor $$\ln(n)$$ inside the latter, and

split it into two products, then factor $$~\displaystyle\prod_{n=2}^N\ln(n)~$$ in both sides, and you'll get $$P_1(1-P_2)$$, whose

asymptotics you'll then have to evaluate, which you'll do by assessing the order of both $$P_1$$ and

$$P_2$$ individually. This will be done by taking their logarithms, and employing an estimation

based on integral approximation. Be warned, this will probably involve special functions, such

as the logarithmic or exponential integral, for instance. The limit ultimately diverges.