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\section*{Some problems for the DVI class}

\begin{enumerate}
\item
Let $f$ be a real-valued, nonnegative function on the positive half-axis.
For every positive integer $i$, put $a_i := f(i)$.
Assume that $f$ is monotonically decreasing. Then
$\sum_{i=1}^{\infty} a_i$ is finite iff $\int_1^{\infty} f(x)\,dx$ is.

\item
And here's a harder one: consider nonnegative solutions $\epsilon$,
$\delta$ of the equation
\[
\bigg(1 + \epsilon \Big(1 + \epsilon \big(1 + \epsilon (1 + \epsilon
    ) \big) \Big) \bigg) =
\sqrt{1 + \delta \sqrt {1 + \delta \sqrt {1 + \delta \sqrt{1 + \delta}}}}
\]
Is it true (and if so, in which sense) that $\epsilon \ll \delta$ as
$\delta \rightarrow 0$ ? What happens if the equation is changed to
\[
\bigg(1 + \epsilon \Big(1 + \epsilon \big(1 + \epsilon (1 + \epsilon
    )^2 \big)^2 \Big)^2 \bigg)^2 =
\sqrt[4]{1 + \delta \sqrt[4] {1 + \delta \sqrt[4] {1 + \delta \sqrt[4]{1 + \delta}}}}
\]

\item
Prove that the series
\[
\sum_{n=1}^{\infty} \frac{(-1)^n\,n}{(n+1)^2}
\]
does converge. (Hint: don't try to use problem 1).
\end{enumerate}
Solutions have to be submitted by yesterday, 0730~AM at my office in
ascii, latin1 and utf-8 encoding.

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