There are two problems with that step.Bashyboy said:So:
-1 < x^2 < 1
+/- sqrt(-1) < x < +/- sqrt(1)
Bashyboy said:Would it be (-1, 1)?
A power series representation is an infinite series of the form ∑n=0^∞ an(x-c)n, where an are coefficients, x is the variable, and c is the center of the series. This representation is used to approximate functions with polynomials and can be used to find values of the function at any point within its radius of convergence.
The radius of convergence is the distance from the center of the power series representation in which the series converges. It is typically denoted by R and can be found by using the ratio test or the root test.
A power series representation is used to approximate functions by finding the sum of an infinite number of terms of the series. By increasing the number of terms used, the accuracy of the approximation increases. This is especially useful for functions that are difficult to evaluate directly.
A Taylor series is a power series representation centered at any point, while a Maclaurin series is a power series representation centered at 0. This means that the coefficients of a Maclaurin series have a specific formula that can be used to find them, while the coefficients of a Taylor series may need to be found using derivatives of the function.
A power series representation is closely related to the derivatives of a function through its coefficients. The coefficients of a power series can be found by taking derivatives of the function at the center of the series. This means that the power series representation can be used to find the derivatives of a function at any point within its radius of convergence.