91影视

Find Maclaurin series for the given pairs of functions, using these steps: (a) Use substitution in the appropriate Maclaurin series to find the Maclaurin series for the given function. (b) Use Theorem 8.11 and your answer from part (a) to find the Maclaurin series for the given function. (c) Find the Maclaurin series for the function in (b), using multiplication and substitution with the appropriate Maclaurin series. Compare your answers from (b) and (c).

(a)$e^{-x^2}$

(b)$x{e}^{-x^2}$

Find the interval of convergence for each power series in Exercises 21鈥�48. If the interval of convergence is finite, be sure to analyze the convergence at the endpoints.

$\underset{k=0}{\overset{鈭�}{鈭�}}\frac{1}{3^k+5^k}{x}^k$

In Exercises 31鈥�40 find the Maclaurin series for the specified function. Note: These are the same functions as in Exercises 21鈥�30.

$f\left(x\right)=\ln (1+x)$

Find Maclaurin series for the given pairs of functions, using these steps: (a) Use substitution in the appropriate Maclaurin series to find the Maclaurin series for the given function. (b) Use Theorem 8.11 and your answer from part (a) to find the Maclaurin series for the given function. (c) Find the Maclaurin series for the function in (b), using multiplication and substitution with the appropriate Maclaurin series. Compare your answers from (b) and (c).

(a) ${\tan }^{-1}\left(\frac{x^2}{3}\right)$

(b)$\left(\frac{x}{9+x^4}\right)$

In Exercises 31鈥�34 in Section 8.2 you were asked to find the Maclaurin series for the specified function. Now find the Lagrange鈥檚 form for the remainder $R_n\left(x\right)$, and show that $\underset{n鈫�鈭�}{\lim }{R}_n\left(x\right)=0$on the specified interval.

$e^x,\mathrm{鈩�}$

ind the Maclaurin series for $e^{-x^2}$, and use it to approximate ${鈭�}_0^1{e}^{-x^2}dx$ to within 0.001 of its value. How many terms would you need to approximate the integral to within${10}^{-6}$ of its value?

Find the interval of convergence for each power series in Exercises 21鈥�48. If the interval of convergence is finite, be sure to analyze the convergence at the endpoints.

$\underset{k=0}{\overset{鈭�}{鈭�}}\frac{{\left(k!\right)}^2}{\left(2k!\right)}{\left(x+2\right)}^k$

Explore the Taylor series for the given pairs of functions, using these steps: (a) Find the Taylor series for the given function at the specified value of x 0 and determine the interval of convergence for the series. (b) Use Theorem 8.11 and your answer from part (a) to find the Taylor series for the given function for the same value of x 0. Also, find the interval of convergence for your series.

(a) $\frac{1}{1-x}$,$x_0=3$

(b)$\frac{1}{{\left(1-x\right)}^2}$

In Exercises 31鈥�40 find the Maclaurin series for the specified function. Note: These are the same functions as in Exercises 21鈥�30.

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

In Exercises 31鈥�34 in Section 8.2 you were asked to find the Maclaurin series for the specified function. Now find the Lagrange鈥檚 form for the remainder $R_n\left(x\right)$, and show that $\underset{n鈫�鈭�}{\lim }{R}_n\left(x\right)=0$on the specified interval.

$\sin x,\mathrm{鈩�}$