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

ln (1.75)

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

cos (0.75)

Let $\Sigma a_n$ be a convergent series, and let

$$R_N=a_{N+1}+a_{N+2}+\cdots \cdot$$

be the remainder of the series after the first N terms. Prove that

$$\lim _{N \rightarrow \infty} R_N=0 .$$

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.

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.

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...