MHB What is the Radius of Convergence for the Power Series?

[Julio1]

Find the radius of convergence of the power series $\displaystyle\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}(z-1)^n}{n+1}$, $z\in \mathbb{C}.$

Spoiler

Hello !, get as ratio this: $R=\dfrac{1}{|z-1|}.$ And this is equal?

 

[Prove It]

Find the ratio of convergence of the power series $\displaystyle\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}(z-1)^n}{n+1}$, $z\in \mathbb{C}.$

Spoiler

Hello !, get as ratio this: $R=\dfrac{1}{|z-1|}.$ And this is equal?

First of all, you are finding a RADIUS of convergence using the ratio test.

You need to set the ratio of the absolute value of two consecutive terms less than 1, and then solve for z.

You are close...

 

[chisigma]

Find the radius of convergence of the power series $\displaystyle\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}(z-1)^n}{n+1}$, $z\in \mathbb{C}.$

Spoiler

Hello !, get as ratio this: $R=\dfrac{1}{|z-1|}.$ And this is equal?

Setting s= z-1 the series becomes...

$\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^{n+1}\ s^{n}}{n+1}\ (1)$

... and applying the ratio test You can verify that is R=1...

Kind regards

$\chi$ $\sigma$