91Ó°ÊÓ

Let f(x) be a function that is continuous, positive, and decreasing on the interval $[1,∞)$such that $\underset{x→∞}{lim}f\left(x\right)=\mathrm{Î\pm }>0$, What can the integral tells us about the series$\underset{k=1}{\overset{∞}{∑}}f\left(k\right)$ ?

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.

$\left\{\frac{k!}{1·3·5···(2k-1)}\right\}$

Explain how to change the index of the series$\underset{k=1}{\overset{∞}{∑}}{a}_k$to start with an initial value other than$1$.

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

If M=0 then$\frac{a_k}{b_k}→$

Explain why the sum of a series satisfying the hypotheses of the alternating series test is between any two consecutive terms in its sequence of partial sums.

Some Convergent Sequences Involving Exponents: For any real number p > 0, the following sequences converge. Fill in each blank with the appropriate value.

$p^{\frac{1}{k}}→$

Give the first five terms of the following recursively defined sequence:

$a_1=1$, and $a_k=a_{k-1}+2$for $kâ‰\yen 2$.

Also, give a closed formula for the sequence.

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.

A series, including the meaning of a term of the series

Provide a more general statement of the limit comparison test in which$\underset{k=1}{\overset{∞}{∑}}{a}_k$and$\underset{k=1}{\overset{∞}{∑}}{b}_k$ are two series whose terms are eventually positive. Explain why your statement is valid.

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 $\{a_k.b_k\}$diverges.