Barrons AP Calculus

BC ONLY Example 17 __ Does the series converge or diverge? SOLUTION: diverges, because it is a p-series ...

the latter is the general term of the divergent p-series , where and . Remember in using the Comparison ...

Since diverges, also diverges by the Limit Comparison Test. THE RATIO TEST Let be a nonnegative ...

But if p > 1 then converges, while if p 1 then diverges. This illustrates the failure of the Ratio Tes ...

Example 26 __ Does the series converge or diverge? SOLUTION: The alternating harmonic series converg ...

Example 28 __ Determine whether converges absolutely, converges conditionally, or diverges. SOLUTION: We see tha ...

Evaluating the sum of the first n terms of an alternating series, given by , yields an app ...

summed to approximate to three decimal places the value of . . . ? SOLUTION: Since and , ...

BC ONLY Example 33 __ Find all x for which the following series converges: SOLUTION: By the Ratio Test, the s ...

SOLUTION: which is less than 1 if |x − 2| < 2, that is, if 0 < x < 4. Series (4) conve ...

Example 38 __ Find the intervals of convergence of the power series for f(x) and f ′(x). SOLUTION: also, and BC ...

in the interval of convergence of the power series for f(x). SOLUTION: Obtain a series for by long divisi ...

is called the Taylor series of the function f about the number a. There is never more than one power se ...

BC ONLY Example 42 __ Find the Maclaurin series for . SOLUTION: Then Note that this agrees exactly with the ...

COMMON MACLAURIN SERIES We list here for reference some frequently used Maclaurin series expansions ...

BC ONLY C4. Approximating Functions with Taylor and Maclaurin Polynomials The function f(x) at the point x ...

(a) (b) BC ONLY In Figure N10–2 we see the graphs of f(x) and of the Taylor polynomials: Figure N10–2 Notice ...

(c) (a) (b) Approximate sin using each of the four polynomials. BC ONLY SOLUTIONS: The derivatives of the sine ...

(c) (a) (b) (c) (a) (b) Figure N10–3b To four decimal places, . Evaluating the polynomials at , we get We see that ...

(c) above, in [0,2.5] × [−1,1]. Figure N10–4 ln . For x = 1.3 the Taylor series converges by the Alternating Seri ...