Module 5: Introduction to Infinite Sequences and Series

In this module, you will be introduced to infinite sequences and series. The module content is split into two lessons.

Project 5:
All sequences in this project are assumed to begin with k=1, unless otherwise noted.
Recall piecewise-defined functions such as
$\inline f(x)=\begin{cases} x\cdot \sin \frac{1}{x} & \text{ if } x \neq 0\\ 0 & \text{ if } x = 0 \end{cases}$

Write out the first 12 terms of the “piecewise-defined sequence”
$\inline a_{k}= f(x)=\begin{cases} \frac{1}{k^{2}} & \text{ if k is odd }\\ \frac{2}{k} & \text{ if k is even } \end{cases}$
Write out the first 12 terms of the “piecewise defined sequence”
$\inline b_{k}= f(x)=\begin{cases} \frac{k-1}{k} & \text{ if k is odd }\\ \frac{1-k}{k} & \text{ if k is even } \end{cases}$
Each of the “pieces” in #1 and #2 are called a subsequence of its respective sequence. Note that each of the four subsequences converges but while the sequence {a_k} in #1 converges to zero, the sequence {a_k} in #2 diverges since the subsequence consisting of the odd-numbered terms converges to 1 and subsequence consisting of the even-numbered terms converges to -1
Find a piecewise formula definition of the sequence 0, 1, 0, 2, 0, 4, 0, 8, 0, 16, 0,…
Do a search on the internet to find some reliable web site or page that states and discusses the Bolzano-Weierstrass Theorem. Write at least one standard page (“typed,” double-spaced, and with no more than 1.25 inch margins) that discusses how the Bolzano-Weierstrass Theorem applies to the sequences you dealt with in #1 and #2, and how the Bolzano-Weirstrass Theorem relates the Bounded Monotonic Sequence Theorem covered in lesson 1 of this module.