Find $\displaystyle \sum_{a=1}^\infty \frac{(-1)^{a+1}(x-1)^a}{a}$ with interval of convergence $(0,2]$ and radius of convergence 1.

**Note:**

1. The link for interval and radius of convergence is here.

2. The $\text{log}(x)$ in the answer section means log base 10.