91Ó°ÊÓ

Show that the radius of convergence for the binomial series is $1$when $p$is not a positive integer. What is the radius of convergence when $p$ is a positive integer? (Hint: Consider Exercise 57.)

Find the Maclaurin series for the functions in Exercises 51–60

by substituting into a known Maclaurin series. Also, give the

interval of convergence for the series.

In exercises 59-62 concern the binomial series to find the maclaurin series for the given function .

$\sqrt[3]{1+x}$

Find the Maclaurin series for the functions in Exercises 51–60

by substituting into a known Maclaurin series. Also, give the

interval of convergence for the series.

Use Theorem 8.12 and the results from Exercises 41–50 to find series equal to the definite integrals in Exercises 51–60.

${∫}_{0.5}^1{\mathrm{x}}^2{\mathrm{e}}^{-3{\mathrm{x}}^2}\mathrm{dx}$

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

Taylor polynomial

Find the interval of convergence of the power series

Let $f(x$) be a function such that the power series in $x-x_0,$$\underset{k-0}{\overset{∞}{∑}}{a}_k{\left(x-x_0\right)}^k$converges absolutely to $f$on the interval I. If $G_1$and $G_2$are two antiderivatives for $f$, explain why the power series in $x-x_0$for $G_1$and $G_2$have the same interval of convergence.

Explain why $\underset{n→∞}{\lim }\frac{{\left|x\right|}^n}{n!}=0$for every value of x.

What is meant by the interval of convergence for a power series in $x$? How is the interval of convergence determined? If a power series in $x$has a nontrivial interval of convergence, what types of intervals are possible?