Activity 8.4.5.
(a)
Explain why the series
\begin{equation*}
1 - \frac{1}{4} - \frac{1}{9} + \frac{1}{16} + \frac{1}{25} + \frac{1}{36} - \frac{1}{49} - \frac{1}{64} - \frac{1}{81} - \frac{1}{100} + \cdots
\end{equation*}
must have a sum that is less than the series
\begin{equation*}
\sum_{k=1}^{\infty} \frac{1}{k^2}\text{.}
\end{equation*}
(b)
Explain why the series
\begin{equation*}
1 - \frac{1}{4} - \frac{1}{9} + \frac{1}{16} + \frac{1}{25} + \frac{1}{36} - \frac{1}{49} - \frac{1}{64} - \frac{1}{81} - \frac{1}{100} + \cdots
\end{equation*}
must have a sum that is greater than the series
\begin{equation*}
\sum_{k=1}^{\infty} -\frac{1}{k^2}\text{.}
\end{equation*}
(c)
Given that the terms in the series
\begin{equation*}
1 - \frac{1}{4} - \frac{1}{9} + \frac{1}{16} + \frac{1}{25} + \frac{1}{36} - \frac{1}{49} - \frac{1}{64} - \frac{1}{81} - \frac{1}{100} + \cdots
\end{equation*}
converge to 0, what do you think the previous two results tell us about the convergence status of this series?
