- #1
aydin
Homework Statement
find the radius of convergence and interval of convergence of the series
∞
Σ (3x-2)^2 / n 3^n
n=1
The formula for finding the radius of convergence for a power series is R = 1 / limn→∞ |an / an+1|, where an is the nth term in the series. In this case, an = (3x-2)^2/n 3^n. To find the radius of convergence, you will need to take the limit as n approaches infinity and then solve for x.
A series has a finite radius of convergence if the limit in the formula for finding the radius of convergence is a positive real number. This means that the series will converge for all values of x within a certain interval. If the limit is equal to infinity, then the series has an infinite radius of convergence and will converge for all values of x.
No, the radius of convergence for a series cannot be negative. The radius of convergence represents the distance from the center of the series to the edge of the interval where the series converges. Since distance cannot be negative, the radius of convergence must be a positive real number or infinity.
The value of n does not have a direct effect on the radius of convergence for this series. The only way it can affect the radius of convergence is if it is part of the nth term, an, which is used to calculate the limit in the formula for the radius of convergence. Otherwise, the value of n does not play a role in determining the radius of convergence.
Yes, it is possible for the interval of convergence to be larger than the radius of convergence. The radius of convergence represents the distance from the center of the series to the edge of the interval where the series converges. However, the series may still converge at points outside of this interval, as long as it is within the interval of convergence. This is why the interval of convergence can be larger than the radius of convergence.