91Ó°ÊÓ

Use Exercise 5 to explain why the ratio test will be inconclusive for every series $\underset{k=1}{\overset{∞}{∑}}{a}_k$in which $a_k$ is a rational function of $k$ .

Explain how you could adapt the limit comparison test to analyze a series $\underset{k=1}{\overset{∞}{∑}}{a}_k$ in which all of the terms are negative.

Explain why a series satisfying the hypotheses of the alternating series test has a sum with the same sign as the first term in the series.

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition or description with a graph or an algebraic example.

The limit of a sequence.

Fill in the blanks to complete each of the following theorem statements.

Basic Limit Rules for Convergent Sequences: If $\left\{a_k\right\}\mathrm{and}\left\{b_k\right\}\mathrm{are}\mathrm{convergent}\mathrm{sequences}\mathrm{with}{a}_k→L\mathrm{and}{b}_k→M\mathrm{as}k→∞$and if c is any constant, then

$a_k{b}_k→$

Dominance Relationships for Sequences: Order the following sequences by dominance when $a>0andb>1$

$\left\{k^a\right\}$

In Exercises 4–11, give examples of sequences satisfying the given conditions or explain why such an example cannot exist.
Two divergent sequences $\left\{a_k\right\}$and $\left\{b_k\right\}$such that the sequence $\{a_k.b_k\}$converges.

Explain why the sequence of partial sums of the series $\underset{k=1}{\overset{∞}{∑}}{a}_k$in which $a_k>0$for every $k∈{\mathrm{ℤ}}^{+}$is strictly increasing.

Limits of sequences: Determine whether the sequences that follow are bounded, monotonic and/or eventually monotonic.

Determine whether each sequence converges or diverges. If the sequence converges, find its limit.

$\left\{\sin \frac{\mathrm{Ï€}}{2k}\right\}$.

Determine whether the series $\frac{99}{40}-\frac{99}{20}+\frac{99}{10}-\frac{99}{5}+…$converges or diverges. Give the sum of the convergent series.