Simplify log expression with infinite series $\log x - \sum_{i=1}^{\infty} \frac{x^i}{i}$

Question

How would I simplify the expression?

$$\log x - \sum_{i=1}^{\infty} \frac{x^i}{i}$$

I'm fairly confident the series is divergent, if its not can you explain how it converges and where to go from there?

Update: The series can be simplified by a simple identity, due to the fact that it's actually a Taylor series expansion.

Answer 1

Hint: taylor series: $${\displaystyle \log(1-x)=-\sum _{i=1}^{\infty }{\frac {x^{i}}{i}}=-x-{\frac {x^{2}}{2}}-{\frac {x^{3}}{3}}-\cdots \quad {\text{ for }}|x|<1}$$