Chapter 8: Power Series

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

power series in x.

The Calculus of Power Series: Let \(\sum_{k=0}^{\infty }a_{k}\left ( x-x_{0} \right )^{k}\) be a power series in \(x-x_{0}\) that converges to a function \(f(x)\) on an interval \(I\).

The derivative of \(f\) is given by \(f^{\prime}\left( x \right )=\)____.

The interval of convergence for this new series is ______, with the possible exception that _____.

Interval of convergence and radius of convergence: Find the interval of convergence and radius of convergence for each of the given power series. If the interval of convergence is finite, test the series for convergence at each of the endpoints of the interval.

$\sum _{k=2}^{\infty }\frac{1}{\ln k}{x}^k$

Find third-order Maclaurin or Taylor polynomial for the given function about the indicated point.

$\tan x,x_0=\frac{\pi }{3}$

If $f$is a function such that $f\left(0\right)=-3$and localid="1650438953513" role="math" $f\text{'}\left(x\right)=2f\left(x\right)$every value of $x$, find the Maclaurin series for $f$.

Show that the power series $\sum _{k=0}^{\mathrm{\infty }} \frac{(-1{)}^k}{2k+1}{x}^{2k+1}$converges conditionally when $x=1$and when $x=-1$. What does this behavior tell you about the interval of convergence for the series?

What is the relationship between a Maclaurin series and a power series in x?

If f(x) is an nth-degree polynomial and $P_n\left(x\right)$ is the nth Taylor polynomial for fat $x_0$, what is the nth remainder $R_n\left(x\right)$? What is $R_{n+1}\left(x\right)$?