Skip to main content\(\newcommand{\dollar}{\$}
\DeclareMathOperator{\erf}{erf}
\DeclareMathOperator{\arctanh}{arctanh}
\DeclareMathOperator{\arcsec}{arcsec}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Activity 8.4.2.
Remember that, by definition, a series converges if and only if its corresponding sequence of partial sums converges.
(a)
Calculate the first few partial sums (to 10 decimal places) of the alternating series
\begin{equation*}
\sum_{k=1}^{\infty} (-1)^{k+1}\frac{1}{k}\text{.}
\end{equation*}
Label each partial sum with the notation \(S_n = \sum_{k=1}^{n} (-1)^{k+1}\frac{1}{k}\) for an appropriate choice of \(n\text{.}\)
(b)
Plot the sequence of partial sums from part (a). What do you notice about this sequence?
