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\title{Sequences and Series}
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\begin{enumerate}
\item Consider the series $\sum_{i=1}^\infty a_i(-1)^{i+1}$ where
for all $i$ we have
$a_i \geq 0$ and the sequence $(a_i)$ converges to $0.$
Prove that the series is convergent.
{\em Hint: Let $s_m = \sum_{i=0}^m a_i.$ Prove that $(s_{2n}$ is monotone
increasing and bounded above by $a_1.$ Do something similar with
$(s_{2n-1}.$ Then apply a theorem to each of these two sequences,
and then stitch the results together somehow}.
\item Consider the sequence $(t_i)$ where
\[ t_m = \sum_{i=1}^m 1/i - \int_{1}^m x^{-1} dx.\]
Show that $t_n > 1/n > 0 \forall n \in {\mathbb N},$ and moreover
that $t_n > t_{n+1} \forall n \in {\mathbb N}.$ Conclude that
the sequence whose $n$-th term is
\[ 1 + 1/2 + 1/3 + \ldots + 1/n - \log n \] is convergent.
(The limit is often called $\gamma,$ or {\em Euler's number} or
{\em Euler's constant}).
\end{enumerate}
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