fixed various typos

integralAndDivergenceTest/exercises/divergenceTestSequenceOrSeries.tex

@@ -18,7 +18,8 @@
 The \emph{sequence} $a_n$:
 
 \begin{multipleChoice}
-\choice[correct]{converges to $0$.}
+\choice{converges to $0$.}
+\choice[correct]{converges to $\frac{2}{3}$.}
 \choice{diverges by the divergence test.}
 \choice{converges by the divergence test.}
 \choice{might converge or diverge; the divergence test is inconclusive.}

limitsOfSequences/exercises/sequenceBoundedMonotonic3.tex

@@ -11,7 +11,7 @@
 
 \begin{document}
 \begin{exercise}
-Consider$\{a_n \}_{n=1}$ where $a_n = 2+ \sin(n\pi)$.  Then, the sequence is:
+Consider$\{a_n \}_{n=1}$ where $a_n = 2+ \sin(n)$.  Then, the sequence is:
 
 \begin{selectAll}
 \choice{increasing}

sumsOfSequences/digInWhatIsASeries.tex

@@ -336,14 +336,9 @@ \section{Properties of sums}
 \begin{explanation}
 Note that $s_3 = a_1+a_2+a_3$ and $s_1 = a_1$.  Thus, 
 
-\begin{tabular}{rl}
-$s_3$ &= $\cancel{a_1}+a_2+a_3$\\
-$-(~ s_1$ &= $\cancel{a_1} ~)$\\
-\hline
-$s_3-s_1$ &= $a_2+a_3$\\
-\end{tabular}
-
-Using the formula for $s_n$ gives $s_3 = \answer[given]{1}$ and $s_1=\answer[given]{-1}$, so $a_1+a_2 =  \answer[given]{2}$.
+\[s_3 -s_1 = \left(\cancel{a_1}+ a_2+a_3\right) -\cancel{a_1} = a_2+a_3\]
+
+Using the formula for $s_n$ gives $s_3 = \answer[given]{1}$ and $s_1=\answer[given]{-1}$, so $a_2+a_3 =  \answer[given]{2}$.
 \end{explanation}
 \end{question}
 

sumsOfSequences/exercises/seriesTelescoping2.tex

@@ -17,7 +17,7 @@
 \sum_{k=1}^{\infty} \ln \left(\frac{k}{k+1}\right)
 \]
 
-Let $s_n = \sum_{k=1}^{\infty}$.  An explicit formula for $s_n$ for $n \geq 1$ is:
+Let $s_n = \sum_{k=1}^n a_k$.  An explicit formula for $s_n$ for $n \geq 1$ is:
 
 \[
 s_n = \answer{-\ln(n+1)}
@@ -32,7 +32,7 @@
 \end{multipleChoice}
 
 \begin{hint}
-To determine whether the series converges or diverges, complete the following argument:
+To determine whether the series converges or diverges, complete the following argument.
 
 The first step to try to determine if a series $\sum_{k=1}^{\infty}a_k$ converges is to consider the sequence of partial sums, defined by $s_n = \sum_{k=1}^{n} a_k = a_1+\ldots + a_n$ for $n \geq 1$, and to try to find an \emph{explicit} formula for $s_n$.