1.
Determine whether each of the following series converges. If it converges, find its sum.
a) \(\sum_{n=0}^\infty (2/3)^n\)\(\qquad\) b) \(\sum_{n=0}^\infty (-1/4)^n\)\(\qquad\) c) \(\sum_{n=0}^\infty (-2)^n\)
Exercises 3.4 Exercises
2.
Use the comparison test to prove that
\begin{equation*}
\sum_{n=1}^\infty \frac{1}{n^2+1}
\end{equation*}
converges.
3.
Determine whether each series converges or diverges.
a) \(\sum_{n=1}^\infty 1/\sqrt{n}\)\(\qquad\) b) \(\sum_{n=1}^\infty 1/n^{3/2}\)\(\qquad\) c) \(\sum_{n=1}^\infty 1/n^{-2}\)
4.
Use the limit comparison test to determine whether
\begin{equation*}
\sum_{n=2}^\infty \frac{1}{n^3-1}
\end{equation*}
converges.
5.
Determine whether each series is absolutely convergent, conditionally convergent, or divergent.
a) \(\sum_{n=1}^\infty (-1)^{n+1}/\sqrt{n}\)\(\qquad\) b) \(\sum_{n=1}^\infty (-1)^n/n^2\)\(\qquad\) c) \(\sum_{n=1}^\infty (-1)^n\)
6.
Use the ratio test to prove that
\begin{equation*}
\sum_{n=1}^\infty \frac{n^2}{2^n}
\end{equation*}
converges absolutely.
7.
Use the ratio test to show that
\begin{equation*}
\sum_{n=1}^\infty \frac{n!}{3^n}
\end{equation*}
diverges.
8.
Use the root test to determine whether
\begin{equation*}
\sum_{n=1}^\infty \left(\frac{3n}{4n+1}\right)^n
\end{equation*}
converges.
9.
Give two series \(\sum x_n\) and \(\sum y_n\) such that \(\lim |x_n|^{1/n}=1=\lim |y_n|^{1/n}\text{,}\) but \(\sum x_n\) converges and \(\sum y_n\) diverges.
10.
Let
\begin{equation*}
s=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}.
\end{equation*}
Find \(N\) such that
\begin{equation*}
\left|s-\sum_{n=1}^N \frac{(-1)^{n+1}}{n}\right| \lt 0.01.
\end{equation*}
11.
Let \(|x|<1\text{.}\) Compute the Cauchy product of \(\sum_{n=0}^\infty x^n\) with itself and use Theorem 3.3.1 to prove that
\begin{equation*}
\sum_{n=0}^\infty (n+1)x^n=\frac{1}{(1-x)^2}.
\end{equation*}
12.
Use the limit form of Raabe’s test (Corollary 3.3.5) to give another proof that
\begin{equation*}
\sum_{n=1}^\infty \frac{1}{n^p}
\end{equation*}
converges for \(p>1\) and diverges for \(p<1\text{.}\) Explain why the case \(p=1\) is inconclusive for Raabe’s test.
