jkh4 said:is it possible for "R" (radius of convergence) to be negative?
for example: -|x|<1 and R=-1?
Yes, it is possible for the radius of convergence to be negative for a power series. This means that the series will not converge for any value of x and therefore does not represent a valid function.
A negative radius of convergence indicates that the power series diverges for all values of x. This means that the series does not have a defined limit as x approaches a certain value.
The radius of convergence is determined by using the ratio test on the coefficients of the power series. If the limit of the ratio is a finite number, then the radius of convergence is the reciprocal of that number. If the limit is infinity, then the radius of convergence is zero. If the limit is zero, then the radius of convergence is infinity.
No, a power series cannot have a negative radius of convergence and still converge. A negative radius of convergence means that the series diverges for all values of x and therefore does not have a valid limit.
A negative radius of convergence means that the series diverges for all values of x, so the interval of convergence is empty. This means that the power series does not converge for any value of x and does not represent a valid function.