Series for log(z) for absolute value of (z-1) less than 1

Theorem

The following formula holds for $|z-1| \leq 1$ and $z \neq 0$: $$\log(z) = -\displaystyle\sum_{k=1}^{\infty} \dfrac{(-1)^k(z-1)^k}{k},$$ where $\log(z)$ denotes the logarithm.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $4.1.26$