Section 11.10 B

Key Problems: 23, 25, 28, 30, 31, 39-43, 55, 56, 60, 63

Problem 23

Find the Taylor series for $f(x)$ centered at the given value of a. [Assume that f has a power series expansion. Do not show that $R_n(x)\to 0$.] Also find the associated radius of convergence.

$$f(x)=\ln x,a=2$$

Problem 25

$$f(x)=e^{2x},a=3$$

Problem 28

$$f(x)=\cos x,a=\pi /2$$

Problem 30

$$f(x)=\sqrt{x},a=16$$

Problem 31

Prove that the series obtained in Exercise 13 represents $\cos x$ for all $x$.

Problem 40

Use a Maclaurin series in Table 1 to obtain the Maclaurin series for the given function.

$$f(x)=\sin (\pi x/4)$$

Problem 41

$$f(x)=2\cos 2x$$

Problem 42

$$f(x)=e^{3x}-e^{2x}$$

Problem 43

$$f(x)=x\cos \left(\frac{1}{2}{x}^2\right)$$

Problem 55

Use the Maclaurin series for $\cos x$ to compute $\cos 5°$ correct to five decimal places.

Problem 56

Use the Maclaurin series for $e^x$ to calculate $1/\sqrt[10]{e}$ correct to five decimal places.

Problem 60

Evaluate the indefinite integral as an infinite series.

$$\int x^2\sin (x^2)dx$$

Problem 63

Use series to approximate the definite integral to within the indicated accuracy.

$${\int }_0^{1/2}{x}^3\arctan xdx\text{(four decimal places)}$$