Here it looks like you flipped the wrong fraction.ReidMerrill said:Homework Statement
Find all values of x such that the given series would converge
Σ6n(x-5)n(n+1)/(n+11)
Homework Equations
The Attempt at a Solution
By doing the ratio test I found that
lim 6n(x-5)n(n+1)/(n+11) * (n+12)/[6n+1(x-5)n+1(n+2)]
n→inf
The radius of convergence is a mathematical concept used in power series to determine the interval in which the series converges. It represents the distance from the center of the series to the nearest point where the series converges.
The radius of convergence is calculated using the ratio test, which compares the absolute value of the ratio between consecutive terms in the series to a limit. If the limit is less than 1, the series converges; if the limit is greater than 1, the series diverges; and if the limit is equal to 1, the test is inconclusive.
The radius of convergence is important because it determines the range of values for which the power series can be used to approximate the function it represents. If a value falls outside of the radius of convergence, the series will not converge and therefore cannot be used to approximate the function at that point.
No, the radius of convergence must always be a positive value. This is because it represents a distance, and distances cannot be negative. However, the interval of convergence can be negative if the series converges for negative values of x.
The radius of convergence is directly related to the convergence of the series. If the series converges at a specific value of x, then that value must fall within the radius of convergence. Similarly, if the series diverges at a specific value of x, then that value must fall outside of the radius of convergence.