Consider the sequence $\left\{A_n\right\}$ whose nth term is given by

$$A_n=P\left(1+\frac{r}{12}\right)^n$$

where P is the principal, $A_n$ is the amount at compound interest after n months, and r is the interest rate compounded annually. Is $\left\{A_n\right\}$ a convergent sequence? Explain.

Match the sequence with the given $n$th term with its graph. [The graphs are labeled (a), (b), (c), and (d).] (c) $a_n=4-\frac{n}{2}$

Write the first five terms of the geometric sequence with the given nth term. $a_n=(-2)^{n-1}$

Which function grows faster as n approaches infinity? Explain. $f(n)=7^{n}$. $g(n)=(n-1) !$

What does it mean for a sequence to be defined recursively?

Use a Taylor polynomial to approximate the function with an error of less than 0.001.

sin 95$^{\circ}$

Show that the series

$$\sum_{n=1}^{\infty} a_n$$

can be written in the telescoping form

$$\sum_{n=1}^{\infty}\left[\left(c-S_{n-1}\right)-\left(c-S_n\right)\right]$$

where $S_0=0$ and $S_n$ is the nth partial sum.

Prove that every decimal with a repeating pattern of digits is a rational number.

Find a sequence that converges to 3/8. (There are many such sequences.) What is the first term in the sequence that is within 0.001 of the limit?

Give an example of a sequence satisfying the given condition or explain why no such sequence exists. (There is more than one correct answer.)

An unbounded sequence that converges to 100.

Give an example of a sequence satisfying the given condition or explain why no such sequence exists. (There is more than one correct answer.)

A sequence that converges to 3/4.

Give an example of a sequence satisfying the given condition or explain why no such sequence exists. (There is more than one correct answer.)

A monotonically increasing bounded sequence that does not converge.

Give an example of a sequence satisfying the given condition or explain why no such sequence exists. (There is more than one correct answer.)

A monotonically increasing sequence that converges to 10.

Prove that 0.75=0.749999...