Question

Find the Maclaurin series for using the definition of a

Maclaurin series. [Assume that has a power series expansion.
Do not show that .] Also find the associated radius of
convergence.
5. f(x) = (1-x)^-2

Answer 1

The Maclaurin series for f(x) = (1-x)^-2 is the sum of (n+1)x^n from n=0 to infinity, and the associated radius of convergence is 1.

Step-by-step explanation:

The student is asking for the Maclaurin series for the function f(x) = (1-x)^-2 and its radius of convergence. To find the Maclaurin series, we need to look at the pattern that the derivatives of this function follow when evaluated at x = 0. The n-th derivative of f(x) at x = 0 is given by f(n)(0) = n!(n+1). Thus, the Maclaurin series for this function can be expressed as:

Summation from n=0 to infinity of (n!(n+1)x^n) / n!

which simplifies to:

Summation from n=0 to infinity of (n+1)x^n

The radius of convergence for this series can be found using the Ratio Test, which implies that the series converges when the absolute value of x is less than 1, hence the radius of convergence is 1.