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

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

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.

Why are the concentrations of solids and liquids omitted from equilibrium expressions?

Consider the sublimation of iodine at 25.0$^{\circ} \mathrm{C}$: $\mathrm{I}_{2}(s) \longrightarrow \mathrm{I}_{2}(g)$ a. Find $\Delta G_{\mathrm{rxn}}^{\circ}$ at 25.0 $^{\circ} \mathrm{C}$ b. Find $\Delta G_{\mathrm{rxn}}^{\circ}$ at 25.0 $^{\circ} \mathrm{C}$ under the following nonstandard conditions: i. $P_{l_2}$ = 1.00 mmHg. ii. $P_{l_2}$ = 0.100 mmHg. c. Explain why iodine spontaneously sublimes in open air at 25.0$^{\circ} \mathrm{C}$.

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?

The data shown here were collected for the first-order reaction: $\mathrm{N}_{2} \mathrm{O}(g) \longrightarrow \mathrm{N}_{2}(g)+\mathrm{O}(g)$ Use an Arrhenius plot to determine the activation barrier and frequency factor for the reaction.

$$\begin{matrix} \text{Temperature (K)} & \text{Rate Constant (1/s)}\\ \text{800} & \ {3.24 \times 10^{-5}}\\ \text{900} & \text{0.00214}\\ \text{1000} & \text{0.0614}\\ \text{1100} & \text{0.955}\\ \end{matrix}$$

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