nickmai123 said:Well, we can't really help you unless you start the process. Do you have any idea where to go from here?
EDIT: If you've got absolutely no idea where to start, you should just Google "radius of convergence series solutions."
nickmai123 said:What does discontinuities of P(x) have to do with anything? A hint would be writing the DE in the form [tex]y^{''}+P(x)y^{'}+Q(x)y=0[/tex] and using a Fuchs' Theorem.
The radius of convergence is a mathematical concept used in power series to determine the interval in which the series will converge. It represents the distance from the center of the series to the points where the series converges.
The radius of convergence is typically calculated using the ratio test, where the limit of the absolute value of the ratio of consecutive terms in the series is taken. If the limit is less than 1, the series converges, and the radius can be calculated using the formula R = 1/L, where L is the limit. If the limit is greater than 1 or does not exist, the series diverges.
The radius of convergence is important because it tells us the values of x for which the power series will converge. This allows us to determine the domain of the function represented by the series and to make approximations for values of x within the radius of convergence.
No, the radius of convergence cannot be negative. It represents a distance and must be a positive value. If the ratio test yields a negative value, it is likely that the series diverges.
If a series has a known radius of convergence, you can plug in values of x within that radius to determine if the series converges or diverges. If the value of x is outside the radius of convergence, the series will definitely diverge. However, if the value of x is within the radius, further testing may be needed to determine convergence.