(BC only) Taylor and Maclaurin series, radius and interval of convergence

Learning objectives

Write the Taylor series of a function about x = a as the sum of f⁽ᵏ⁾(a)(x − a)ᵏ/k!, and the Maclaurin series as the special case centered at zero, identifying the general term from the pattern of the derivatives at the center and expressing the series in sigma notation

Recall the Maclaurin series for eˣ, sin x and cos x, valid for all real x, and for 1/(1 − x), valid for |x| < 1, and use them as building blocks

Generate new series by substituting an expression for x, by multiplying by a power of x, and by differentiating or integrating term by term within the interval of convergence

Find the radius of convergence of a power series with the ratio test, and state the open interval of absolute convergence centered on a

Test each endpoint of that interval separately with an appropriate convergence test, and give the full interval of convergence with correct open or closed brackets

Apply a power series representation to evaluate an indeterminate limit, to approximate a definite integral whose integrand has no elementary antiderivative, or to approximate the solution of a differential equation