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?
The radius of convergence is a mathematical concept used in the study of power series. It represents the distance from the center of a power series to the nearest point where the series converges, or becomes a finite value.
The radius of convergence can be determined using the ratio test, which involves taking the limit of the ratio of consecutive terms in the power series. If this limit is less than 1, then the series converges. If it is greater than 1, the series diverges. If the limit is exactly 1, further analysis is needed to determine the convergence or divergence of the series.
The radius of convergence is important because it tells us the region in which a power series is valid. If a value falls within the radius of convergence, the series will converge to a finite value at that point. If a value falls outside of the radius of convergence, the series will diverge and have no meaningful value. This concept is especially useful in applied mathematics and physics, where power series are commonly used to approximate functions.
No, the radius of convergence cannot be negative. It represents a distance and therefore must be a positive value. However, it can be infinite if the power series converges for all values of the variable.
The radius of convergence and the interval of convergence are closely related. The interval of convergence is the set of all values of the variable for which the power series converges. The radius of convergence gives the center of this interval. For example, if the radius of convergence is 3, then the interval of convergence is the set of all values within a distance of 3 from the center of the power series.