Property of radius of convergence

In summary, the property of radius of convergence is a mathematical concept used in power series analysis that refers to the maximum distance from the center of a series to the nearest point where the series converges. The radius of convergence is determined using the ratio test, and its significance lies in determining the domain of the series and its validity for approximating a function. The radius of convergence cannot be negative and can be affected by coefficients, powers, and the presence of singularities or asymptotes within the series.
  • #1
librastar
15
0
I have a question regarding the radius of convergence and hopely someone can help me with it.

Suppose [tex]\Sigma[/tex]NANZN-1 is given and if its primitive exists, will these two polynomials have the same radius of convergence?
 
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  • #3
LCKurtz said:
Yes.

Thank you very much.
 

What is the property of radius of convergence?

The property of radius of convergence is a mathematical concept used in power series analysis. It refers to the distance from the center of a power series to the nearest point where the series converges. In other words, it is the maximum value of the variable for which the series will converge.

How is the radius of convergence determined?

The radius of convergence is typically determined by using the ratio test, which compares the absolute value of consecutive terms in the power series. If the limit of this ratio is less than 1, the series will converge and the radius of convergence can be calculated. If the limit is greater than 1, the series will diverge and the radius of convergence is 0.

What is the significance of the radius of convergence?

The radius of convergence is important because it determines the domain of the power series. All values within the radius of convergence will result in a convergent series, while values outside the radius will result in a divergent series. This allows us to determine where the series is valid and can be used to approximate a function.

Can the radius of convergence be negative?

No, the radius of convergence must be a positive value. This is because the radius is a measure of the distance from the center of the series, and distance is always positive. A negative radius would not make sense in this context.

What factors can affect the radius of convergence?

The radius of convergence can be affected by the coefficients of the power series, as well as the variable being raised to a power. In general, larger coefficients and higher powers will result in a smaller radius of convergence. Additionally, the presence of singularities or asymptotes within the series can also impact the radius of convergence.

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