Given that

$$\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}$$

with convergence in (−1, 1), find the power series for the function with the given center a, and identify its interval of convergence. $f(x)=\frac{1}{1-2 x} ; a=0$

I have 100 items of a product in stock. The probability mass function for the product’s demand D is $P(\mathrm{D}=90)=P(\mathrm{D}=100)=P(\mathrm{D}=110)=\frac{1}{3}$. a. Find the mass function, mean, and variance of the number of items sold. b. Find the mass function, mean, and variance of the amount of demand that will be unfilled because of lack of stock.

Express each repeating decimal as a fraction. (a)

$$5 . \overline { 23 }$$

(b)

$$1.3 \overline { 7 }$$

(c)

$$2.1 \overline { 35 }$$

For two invertible $n \times n$
matrices $A$ and $B$, determine the formulas stated below is necessarily true:

$A + B$ is invertible, and$(A + B)^{-1} = A^{-1} + B^{-1}$

Let $L$ be the set of all strings, including the null string, that can be constructed by repeated application of the following rules: If $\alpha\in L$, then $a\alpha b\in L$ and $b\alpha a\in L$. If $\alpha\in L$ and $\beta\in L$, then $\alpha\beta\in L$.

Prove that if $\alpha$ has equal number of $a$'s and $b$'s, then $\alpha\in L$.

Does the graph of $- 8 x ^ { 2 } - 12 = 3 - y$ open upward or downward?

Suppose that p(x) is the density function for heights of American men, in inches. What is the meaning of the statement p(68)=0.02?

Graph the function defined by $f(x)=\frac{-2}{x+1}$. Give the equation of its vertical asymptote.

Which lines are parallel? How do you know?

Use Figure the periodic table to list all the (a) inert gases, (b) alkalis, (c) halogens, and (d) alkaline earths.

A $20-\text{lb}$ weight is attached to the lower end of a coil spring suspended from a beam and comes to rest in its equilibrium position. The weight is then pulled down $1$ ft below its equilibrium position and released at $t=0$ with an initial velocity of $v_0\,\,\text{ft}/\text{sec}$, directed downward. Neglect the resistance of the medium. Determine the spring constant $k$ and the initial velocity $v_0$ if the amplitude of the resulting motion is $\sqrt{17}/4$ and the period is $\pi/4$.

A radio tower 500 feet high is located on the side of a hill with an inclination to the horizontal of $5^{\circ}$. How long should two guy wires be if they are to connect to the top of the tower and be secured at two points 100 feet directly above and directly below the base of the tower?

Choose the letter for the best answer. Adam and his two brothers went to the zoo. Each ticket to enter the zoo costs $7. Adam bought two bags of peanuts for$4 each, and one of his brothers bought a lion poster for $12. Which expression shows how much money they spent at the zoo in all? A. 7 + 4 + 12 B.$7 \times 3 + 4 + 12$C.$7 \times 3 + 4 \times 2 + 12$D.$( 7 \times 3 ) + ( 4 \times 12 )$

Find the indicated number of elements by referring to the following table of enrollments in a finite mathematics class: $$ \begin{matrix} \text{} & \text{Freshmen} & \text{Sophomores}\\ \text{Arts & Sciences} & \text{19} & \text{14}\\ \text{Business} & \text{66} & \text{21}\\ \end{matrix} $$ Let the universal set U be the set of all 120 students in the class, A the set of students from the College of Arts & Sciences, B the set of students from the College of Business, F the set of freshmen, and S the set of sophomores. $n(A)$