91Ó°ÊÓ

Let$\underset{k=0}{\overset{∞}{∑}}c{r}^k$and$\underset{k=0}{\overset{∞}{∑}}b{v}^k$ be two convergent geometric series. If b and v are both nonzero, prove that $\underset{k=0}{\overset{∞}{∑}}\frac{c{r}^k}{b{v}^k}$ is a geometric series. What condition(s) must be met for this series to converge?

Suppose you invest $100.00 in a bank that pays you 5% interest compounded annually. The balance in the account after k years is given by $a_k=100{(1-0.05)}^k$To the nearest cent, determine the first five terms of the sequence, starting at k = 0. What does k = 0 mean in practical terms ?Determine whether the sequence is bounded. Determine whether the sequence is increasing, decreasing, or not monotonic.

Give a recursive definition for the sequence $1,2,3,4,....$of positive integers. (Hint: Let $a_1=1$.)

Let $\underset{k=1}{\overset{∞}{∑}}{a}_k$be a series in which all the terms are positive. If$\underset{k→∞}{\lim }\frac{a_{k+1}}{a_k}>1$, explain why both the ratio test and the divergence test could be used to show that the series diverges .

What condition(s) must a series $\underset{k=1}{\overset{∞}{∑}}{a}_k$satisfy in order for the series to be absolutely convergent?

Provide a more general statement of the integral test in which the function f is continuous and eventually positive, and decreasing. Explain why your statement is valid.

For the series$\underset{k=0}{\overset{\mathrm{∞}}{∑}} \left(\frac{1}{k+3}−\frac{1}{k+4}\right)$that follow,

Part (a): Provide the first five terms in the sequence of partial sums $\left\{S_k\right\}$.

Part (b): Provide a closed formula for $S_k$.

Part (c): Find the sum of the series by evaluating$\underset{k→∞}{\lim }{S}_k$.

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 sum of a convergent series

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

${\left(\frac{1}{k}\right)}^p→$

If you suspect that a series $\underset{k=1}{\overset{∞}{∑}}{a}_k$ converges, explain why you would want to compare the series with a convergent series, using either the comparison test or the limit comparison test.