Power Series Interval of Convergence

In summary, a power series interval of convergence is the range of values for which a power series will converge or have a finite sum. The radius of convergence can be found using the ratio test, which involves taking the limit of the absolute value of the ratio of the (n+1)th term to the nth term. The convergence of the series may be different at the endpoints of the interval, and the radius of convergence must always be a positive value. The coefficients in a power series can affect its interval of convergence, but they are not the sole determining factor.
  • #1
harrietstowe
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Homework Statement


I need a power series with a radius = pi. (So when you do the ratio test on this power series you get pi)


Homework Equations





The Attempt at a Solution


I tried x^n*sin(n) and thought of stuff like that but couldn't come up with a working power series
 
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  • #2
Look at the ratio or root test and try to figure out how to change the radius of convergence of a power series, so that you can start with a power series whose radius of convergence you know, and turn it into one that solves your problem.
 

1. What is a power series interval of convergence?

A power series interval of convergence is the range of values for which a power series, which is an infinite sum of terms involving a variable raised to increasing powers, will converge or have a finite sum. It is usually denoted by an interval, such as (-R, R), where R is the radius of convergence.

2. How do you find the radius of convergence for a power series?

The radius of convergence can be found by using the ratio test, which involves taking the limit as n approaches infinity of the absolute value of the ratio of the (n+1)th term to the nth term in the power series. If this limit is less than 1, then the series will converge, and the radius of convergence is equal to the value where the limit is taken. If the limit is greater than 1, the series will diverge, and if it is equal to 1, further tests are needed to determine convergence.

3. What happens at the endpoints of the interval of convergence?

At the endpoints of the interval of convergence, the convergence of the series may be different. For example, if the interval is (-R, R), the series may converge at one endpoint and diverge at the other. This can be determined by substituting in the value of the endpoint into the power series and seeing if it converges or diverges.

4. Can the radius of convergence be negative?

No, the radius of convergence must always be a positive value. This is because it represents the distance from the center of the power series where the series will converge. A negative value would not make sense in this context.

5. How is the power series interval of convergence affected by the coefficients?

The coefficients in a power series can affect its interval of convergence. Generally, if the coefficients are larger, the radius of convergence will be smaller, and vice versa. Additionally, if the coefficients follow a specific pattern, such as decreasing in size, the series may have a larger interval of convergence. However, the coefficients alone do not determine the interval of convergence, as it also depends on the variable and the powers it is raised to in the series.

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