Barrons AP Calculus - David Bock
earlier, this is a necessary condition for convergence, but may not be sufficient. We now examine the sum using the same techniq ...
SOLUTION: by an application of L’Hôpital’s Rule. Thus converges. THE p-SERIES TEST A p-series converges if p > 1, but diverge ...
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 < ...
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