The radius of convergence for the power series $\displaystyle\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}(z-1)^n}{n+1}$ is determined using the ratio test, resulting in a radius of convergence R=1. By substituting s = z - 1, the series simplifies to $\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^{n+1}\ s^{n}}{n+1}$. The ratio of the absolute values of consecutive terms must be less than 1 to find the convergence criteria.
PREREQUISITESMathematicians, students studying complex analysis, and anyone interested in understanding power series and their convergence properties.
Julio said:Find the ratio of convergence of the power series $\displaystyle\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}(z-1)^n}{n+1}$, $z\in \mathbb{C}.$Hello !, get as ratio this: $R=\dfrac{1}{|z-1|}.$ And this is equal?
Julio said:Find the radius of convergence of the power series $\displaystyle\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}(z-1)^n}{n+1}$, $z\in \mathbb{C}.$Hello !, get as ratio this: $R=\dfrac{1}{|z-1|}.$ And this is equal?