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. Find the first ten terms of the sequence if P=$9000 and r=0.115.

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...

Consider making monthly deposits of P dollars in a savings account at an annual interest rate r. Use the results of Exercise 76 to find the balance A after t years if the interest is compounded (a) monthly and (b) continuously.

P=$20, r=6%, t=50 years

Use power series to find the limit (if it exists). Verify the result by using L'Hopital's Rule.

$$\lim _{x \rightarrow 0} \frac{\arcsin x}{x}$$

(a) use the mentioned theorem to show that the sequence with the given nth term converges and

(b) use a graphing utility to graph the first ten terms of the sequence and find its limit.

$$a_n=4+\frac{1}{2^n}$$

In the following exercise, match the series with the graph of its sequence of partial sums.

$$\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n-1} 4}{(2 n) !}$$

Find a power series for the function, centered at c, and determine the interval of convergence.

f(x)=3 / 2x-1, c=2

Verify that the infinite series diverges.

$$\sum_{n=1}^{\infty} \frac{n}{2 n+3}=\frac{1}{5}+\frac{2}{7}+\frac{3}{9}+\frac{4}{11}+\cdots$$