# Show that the Taylor series for the principal part of $\log(1+z)$ converges absolutely for $|z|\le1$

Show that the Taylor series for the principal part of $\log(1+z)$ converges absolutely for $|z|\le1$

the Taylor series for

$$\log(1+z)=z-\frac{z^2}{2}+\frac{z^3}{3}-\frac{z^4}{4}+......\sum _{n=1}^\infty\frac{(-1)^{n+1}z^{n}}{n}$$

how to prove the principal part converges absolutely for $|z|\le 1$

Nobody can prove that because it is false. At $z=-1$, the series diverges, since it is equal to$$-1-\frac12-\frac13-\frac14-\cdots$$
Note that $f(z)=\log(1+z)$ is analytic for $|z|<1$ (why?) and not analytic at $z=-1$. So, the radius of convergence of $f$ is $R=1$.
We can easily see that, thus, the principal part of the Laurent series converges uniformly on $|z|<1$.