Barrons AP Calculus - David Bock

SOLUTION: diverges, since the latter is the general term of the divergent p-series where and Remember in using the Comparison Te ...

EXAMPLE 23 Does converge or diverge? SOLUTION: and (See §E2.) Since e > 1, diverges by the Ratio Test. EXAMPLE 24 If the Rati ...

THE ALTERNATING SERIES TEST An alternating series converges if: (1) an + 1 < an for all n, and (2) EXAMPLE 26 Does the series ...

divergent. EXAMPLE 29 Determine whether converges absolutely, converges conditionally, or diverges. SOLUTION: Note that, since t ...

see that the error is always less than that next term. Since |Rn | < an + 1, the error bound for an alternating series is the ...

The set of all values of x for which a power series converges is called its interval of convergence. To find the interval of con ...

diverges if |x − 2| > 2, that is, if x < 0 or x > 4. When x = 0, (4) is 1 − 1 + 1 − 1 + · · · and diverges. When x = 4, ...

and Hence, the power series for f converges if −1 x 1. For the derivative also, and Hence, the power series for f ′ converges if ...

Since when x = 0 we see that c = 1, we have Note that this is a geometric series with ratio r = x and with a = 1; if |x| < 1, ...

SOLUTION: Thus, EXAMPLE 42 Find the Maclaurin series for SOLUTION: Then Note that this agrees exactly with the power series in x ...

Then COMMON MACLAURIN SERIES We list here for reference some frequently used Maclaurin series expansions together with their int ...

C4. Approximating Functions with Taylor and Maclaurin Polynomials. The function f (x) at the point x = a is approximated by a Ta ...

This approximation of e−0.25 is correct to four places. In Figure N10–2 we see the graphs of f (x) and of the Taylor polynomials ...

From the table we know that (b) Figure N10–3a shows the graphs of sin x and the four polynomials. In Figure N10–3b we see graphs ...

SOLUTIONS: (a) The derivatives of ln x at x = 1 are given in the table: From the table we have (b) Figure N10–4 shows the graphs ...

For x > 0, this is an alternating series with terms decreasing in magnitude and approaching 0, so the error committed by usin ...

EXAMPLE 49 Find the Maclaurin series for ln (1 + x) and the associated Lagrange error bound. SOLUTION: Then where the Lagrange e ...

Thus Since this is not an alternating series for x = 0.1, we must use the Lagrange error bound: where x = 0.1 and 0 < c < ...

so correct to four decimal places. EXAMPLE 52 Estimate the error if the approximate formula is used and |x| < 0.02. SOLUTION: ...

EXAMPLE 55 EXAMPLE 56 Show how a series may be used to evaluate π. SOLUTION: Since a series for tan−1 x may prove helpful. Note ...