91Ó°ÊÓ

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

$\text{48.}\sin 3x,\frac{\mathrm{Ï€}}{6}$

In Exercises 41–50, 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

$\left(a\right)f\left(x\right)=x^3\cos \left(\frac{x}{2}\right)\left(b\right)F\left(0\right)=-4$

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 $\underset{n→∞}{\lim }{R}_n\left(x\right)=0$on the specified interval.

$\sqrt{x},1,(1/2,3/2)$

Find the radius of convergence for the given series:$\underset{k=0}{\overset{∞}{∑}}\frac{k!}{\left(k+m\right)!}{x}^k$

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 $\underset{n→∞}{\lim }{R}_n\left(x\right)=0$on the specified interval.

localid="1649857668603" $\ln \left(x\right),3,(2,4)$

In Exercises 41–50, 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 $\mathrm{F}=∫\mathrm{f}$that satisfies the specified initial condition

localid="1650724626739" $\left(\mathrm{a}\right)\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^2{\mathrm{e}}^{-3{\mathrm{x}}^2}\left(\mathrm{b}\right)\mathrm{F}\left(0\right)=1$

In Exercises 49–56 find the Taylor series for the specified function and the given value of $x_0$. Note: These are the same functions and values as in Exercises 41–48.

role="math" localid="1649406173168" $49.\cos \left(x\right),\frac{\mathrm{Ï€}}{2}$

Give precise mathematical definitions or descriptions of the concept that follows. Then illustrate the definition or description with a graph or an algebraic example.

Maclaurin polynomial

Let $f\left(x\right)=\underset{k=0}{\overset{∞}{∑}}{a}_k{\left(x-x_0\right)}^k$and let $G$be the antiderivative for $f$with the property that $G\left(x_0\right)=7$. Find the Taylor series in$x_0$ for $G$.

Explain why $\underset{k=0}{\overset{∞}{∑}}{\left(x-k\right)}^k$ is not a power series.