91影视

In Exercises $49-54$in Section $8.2$you were asked to find the Taylor series for the specified function at the given value of $x_0$. In Exercises $45-50$find the Lagrange's 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.

$\cos x,\frac{蟺}{2},\mathrm{鈩�}$

Find the interval of convergence for power series:$\underset{k=0}{\overset{鈭�}{鈭�}}\frac{1}{1.3.5.....\left(2k+1\right)}{x}^k$

In Exercises $49-54$in Section $8.2$you were asked to find the Taylor series for the specified function at the given value of $x_0$. In Exercises $45-50$find the Lagrange's 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.

$e^x,1,\mathrm{鈩�}$

In Exercises 41鈥�48 find the fourth Taylor polynomial $P_4\left(x\right)$for the specified function and the given value of $x_0$

$46.\sqrt[3]{x},1$

In Exercises 41鈥�50, find Maclaurin series for the given pairs of functions, using these steps:

(a) Use substitution and/or multiplication and the appropriate Maclaurin series to find the Maclaurin series for the given function f .

(b) Use Theorem 8.12 and your answer from part (a) to find the Maclaurin series for the antiderivative that satisfies the specified initial condition

$\left(a\right)f\left(x\right)=\ln (4+x^2)\left(b\right)F\left(0\right)=-2$

Find the interval of convergence for power series:$\underset{k=0}{\overset{鈭�}{鈭�}}\frac{1}{k^k}{\left(x-3\right)}^k$

In Exercises $45-50$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.

$\sin x,蟺,\mathrm{鈩�}$

In Exercises 41鈥�48 find the fourth Taylor polynomial $P_4\left(x\right)$for the specified function and the given value of $x_0$.

$\text{47.}\cos 2x,\frac{蟺}{4}$

In Exercises 41鈥�50, find Maclaurin series for the given pairs of functions, using these steps:

(a) Use substitution and/or multiplication and the appropriate Maclaurin series to find the Maclaurin series for the given function f .

(b) Use Theorem 8.12 and your answer from part (a) to find the Maclaurin series for the antiderivative $F=鈭�f$that satisfies the specified initial condition

$\left(a\right)f\left(x\right)=x^2\sin \left(5x^2\right)\left(b\right)F\left(0\right)=1$

Find the interval of convergence for power series: $\underset{k=0}{\overset{鈭�}{鈭�}}\frac{k^3}{1.3.5....\left(2k+1\right)}{\left(x+1\right)}^k$