1.
Find the radius of convergence and the interval of convergence of each power series.
-
\(\sum_{n=1}^\infty \frac{x^n}{n}\text{,}\)
-
\(\sum_{n=0}^\infty n!x^n\text{,}\)
-
\(\sum_{n=0}^\infty \frac{x^n}{n!}\text{.}\)
Exercises 6.5 Exercises
2.
Find the radius of convergence and the interval of convergence of
\begin{equation*}
\sum_{n=0}^\infty \frac{(x-2)^n}{3^n}.
\end{equation*}
3.
Use the geometric series and term-by-term differentiation to compute
\begin{equation*}
\sum_{n=1}^\infty \frac{n}{3^n}.
\end{equation*}
4.
Starting from the geometric series, derive the power series for \(\ln(1+x)\) and \(\arctan x\) about \(0\text{.}\) In each case, determine the radius of convergence.
5.
Let
\begin{equation*}
f(x)=\begin{cases}
e^{-1/x} & \text{if } x>0,\\
0 & \text{if } x\le 0.
\end{cases}
\end{equation*}
Show that every derivative of \(f\) at \(0\) is equal to \(0\text{.}\) Conclude that \(f\) is not analytic at \(0\text{.}\)
6.
Let \(P_3(x)=x-\frac{x^2}{2}+\frac{x^3}{3}\text{,}\) the third Taylor polynomial of \(\ln(1+x)\) at \(0\text{.}\)
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Compute \(\ln(1+x)-P_3(x)\) in the form given by Taylor’s theorem.
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Use this to show that \(|\ln(3/2)-P_3(1/2)| \le 1/32\text{.}\)
7.
Use the binomial series to write down the first four nonzero terms of the power series for \(\sqrt{1+x}\) about \(0\text{.}\) What is the radius of convergence?
