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Section 8.9 Exploration: Infinite Series
Subsection Interesting Infinite Series
We’ve seen a lot of infinite series examples, and we have a lot of tools for how to think about infinite series. Take a look at some more examples that are a bit strange to think about.
Exploration 8.9.1 . Interesting Infinite Series.
(a)
Consider the series
\begin{equation*}
\sum_{k=1}^\infty \frac{1}{D(k)k^p}
\end{equation*}
where \(D(k)\) is the function that returns the number of digits in the number \(k\) (so \(D(9)=1\) and \(D(435)=3\) ), and \(p\) is a real number.
For what values of
\(p\) does this series converge?
(b)
Consider the series
\begin{equation*}
\sum_{k=1}^\infty \frac{1}{F(k)k^p}
\end{equation*}
where \(F(k)\) is the function that returns the \(k\) th Fibbonacci number, and \(p\) is a real number. As a reminder, the Fibbonacci sequence is:
\begin{equation*}
1,1,2,3,5,8,13,21...
\end{equation*}
where \(F(k)=F(k-1)+F(k-2)\text{.}\)
For what values of
\(p\) does this series converge?
