91影视

In Exercises 21鈥�30 in Section 8.2 you were asked to find the fourth Maclaurin polynomial $P_4\left(x\right)$for the specified function. In Exercises 23鈥�32 we ask you to give Lagrange鈥檚 form for the corresponding remainder, $R_4\left(x\right)$.

localid="1650435468717" $x\sin x$

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.

$\underset{k=1}{\overset{鈭�}{鈭�}}\frac{2^k+1}{\sqrt{k}}{x}^k$

Use the Maclaurin series for cos x to find series representations for$\cos \left(x^3\right),鈭�\cos \left(x^3\right)dx,{鈭�}_0^1\cos \left(x^3\right)dx$

In Exercises 21鈥�30 in Section 8.2 you were asked to find the fourth Maclaurin polynomial $P_4\left(x\right)$for the specified function. In Exercises 23鈥�32 we ask you to give Lagrange鈥檚 form for the corresponding remainder, $R_4\left(x\right)$.

$x^2{e}^x$

Find the Maclaurin series for the specified function:

$e^x$.

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)$\cos \left(4x^3\right)$

(b)$x^2\sin \left(4x^3\right)$

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.

$\underset{k=0}{\overset{鈭�}{鈭�}}{\left(-1\right)}^k\frac{k^2}{3^k}{\left(x-\frac{1}{2}\right)}^k$

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.

33.$\cos x,\mathrm{鈩�}$

Find the Maclaurin series for the specified function:

$\sin \left(x\right)$.

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