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

Question

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 $|z|\le 1$

Possible duplicate of What is the correct radius of convergence for $\ln(1+x)$? — GNUSupporter 8964民主女神地下教會, Dec 10 '17 at 12:31
It should be this one: https://math.stackexchange.com/q/621211/290189 — GNUSupporter 8964民主女神地下教會, Dec 10 '17 at 12:33
Possible duplicate of Uniform and absolute convergence of complex series to $\log(1+z)$ — Martin R, Dec 10 '17 at 12:57
I don't think this question is a duplicate of the questions at those links. — zhw., Dec 10 '17 at 18:02

score 1 · Answer 1 · answered Dec 10 '17 at 17:50

1

Nobody can prove that because it is false. At $z=-1$, the series diverges, since it is equal to$$-1-\frac12-\frac13-\frac14-\cdots$$

answered Dec 10 '17 at 17:50

score 0 · Answer 2 · answered Dec 10 '17 at 12:36

0

Hint:

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 $|z|<1$.

answered Dec 10 '17 at 12:36

2 Answers2