I'm trying to solve the following taylor series $$\sum_{n=0}^\infty \frac{x^n}{n!} \ln(n+1)$$

but the closest I'm getting is that it looks similar to the function, and I'm not sure how else to solve the sum$$xe^x$$

I'm trying to solve the following taylor series $$\sum_{n=0}^\infty \frac{x^n}{n!} \ln(n+1)$$ so I can regularize the following sum $$\sum_{n=1}^\infty \ln(n)$$

Using Borel Regularizaiton I can use the formula $$\sum_{n=1}^\infty \ln(n) = \lim_{t\to1}\mathcal{L}_x\left[\sum_{n=0}^\infty\frac{x^n}{n!}\ln(n+1)\right](t)$$ but I'm unsure on how to solve the Taylor Expansion sum, the closest I've gotten is that the function $xe^x$ looks similar to the expansion.