91影视

Outline the steps you would use to approximate the sum of a convergent series satisfying the hypotheses of the alternating series test to within $$蔚$$ of its value.

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 sequence of partial sums for a given series

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.

$\left\{\frac{1路3路5路路(2k-1)}{3路6路9路路路\left(3k\right)}\right\}$

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

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

If f is a function that is _____ at L, then$f\left(a_k\right)鈫�$

Explain how you could adapt the integral test to analyze a series $\underset{k=1}{\overset{鈭�}{鈭�}}f\left(k\right)$in which the function$f:[1,鈭�)鈫�\mathrm{鈩�}$ is continuous, negative, and increasing.

Use the result of Exercise $79$to approximate the square roots in Exercises $80-83$. In each case, start with $x_0=1$and stop when $\left|x_{k+1}-x_k\right|<0.001.$

$\sqrt{2}$

Whenever a certain ball is dropped, it always rebounds to a height p% (0 < p < 100) of its original position. What is the total distance the ball travels before coming to rest when it is dropped from a height of h meters?

Prove the statements about the convergence or divergence of sequences in Exercises 78鈥�83, referring to theorems in the section as necessary. For each of these statements, assume that r is a real number and p is a positive real number.

If $r=1$then$r^k鈫�1$

Use the result of Exercise 79 to approximate the square roots in Exercises 80鈥�83. In each case, start with $x_0=1$and stop when $\left|x_{k+1}鈭�x_k\right|<0.001$.

81.$\sqrt{3}$

Prove the statements about the convergence or divergence of sequences in Exercises 78鈥�83, referring to theorems in the section as necessary. For each of these statements, assume that r is a real number and p is a positive real number.

If$\left|r\right|>1,$then the sequence$\left\{r^k\right\}$diverges and$\left|r^k\right|鈫�鈭�$