determine whether the series is convergent or divergent. if it is convergent, find its sum. (if the quantity diverges, enter diverges.) [infinity]Σn = 1 1/9+e^-n

Answers

Answer 1

The given series is convergent, and its sum is approximately 0.1524.

How to determine whether the series is convergent or divergent?

To determine whether the series ∑n=1∞ 1/(9+$e^{(-n)}$) is convergent or divergent, we can use the comparison test with the series 1/n.

Since for all n, $e^{(-n)}$ > 0, we have $9 + e^{(-n)}$ > 9, and so $1/(9+e^{(-n)})$ < 1/9.

Now, we can compare the given series with the series ∑n=1∞ 1/9, which is a convergent p-series with p=1.

By the comparison test, since the terms of the given series are smaller than those of the convergent series 1/9, the given series must also converge.

To find the sum of the series, we can use the formula for the sum of an infinite geometric series:

S = a/(1-r)

where a is the first term and r is the common ratio. In this case, the first term is 1/10 (since $e^{(-1)}$ is very small compared to 9.

We can approximate $9+e^{(-n)}$ as 9 for large n), and the common ratio is $e^{(-1)} < 1$. Therefore, the sum of the series is:

S = (1/10)/(1 - $e^{(-1)}$) = (1/10)/(1 - 0.3679) ≈ 0.1524

Therefore, the given series is convergent, and its sum is approximately 0.1524.

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Related Questions

A rectangle has a height 3 times the length of its base. If the base increases by 3 cm and the height by 5 cm then the area is 319 2. What are the dimensions of the rectangle?

Answers

Answer:

The length of a rectangle is 3 times the width.The area is 75 cm^2.

State if the three numbers can be the measures of the sides of a triangle
5, 12, 10
Yes
No

Answers

No it can not since two of the sides of a triangle are the same. Don’t click the links!

Alan ran 3.1 miles in 35 minutes. Courtney ran 3.1 miles 5 minutes faster than Alan. Which number line can be used to find how long it took Courtney to run 3.1 miles?

Answers

Answer:

.................

Step-by-step explanation:

/..........

. Find f(3) for the following function

$f(x)=3(2)^x+6$

a.f(3) = 222
b.f(3) = 9
c.f(3) = 24
d.f(3) = 30

Answers

Answer:

$\boxed {\boxed {\sf D. \ f(3)= 30}}$

Step-by-step explanation:

We are given the function:

$f(x)= 3(2)^x+6$

We want to find f(3), so we must substitute 3 in for x.

$f(3)= 3(2)^3+6$

Solve according to PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction.

Solve the exponent first.

2³= 2*2*2= 4*2= 8

$f(3)= 3(8)+6$

Multiply 8 and 3.

$f(3)= 24+6$

Add.

$f(3)=30$

For the function given, f(3) is 30.

D is the correct answer

Which of the following is the graph of f (x) = x²

Answers

Answer:

Step-by-step explanation:

Which of the following is the graph of f (x) = x

Answer:

whichever one looks like a U opening upward. the bottom of the U should be at the point (0,0) where the x and y axis meet

The members of a student council held a car wash to earn money for a field trip. The equations represent how they calculated their earnings.

5c+8v=234
c=2v
If c
represents the number of cars and v
represents the number of vans, which pair of sentences best describe the results of the fundraiser?

Answers

There are 26 automobiles and 13 vans available for the field excursion.

What is equation?

An algebraic equation, also known as a polynomial equation, is a mathematical equation of the form P=0, where P is a polynomial with coefficients in some field, most commonly the field of rational numbers. A mathematical statement that expresses the connection between two values is simply characterized as an equation. In an equation, the two values are usually equated by an equal sign. Linear equations are first-order equations. Lines in the coordinate system are determined by linear equations. A linear equation in one variable is defined as an equation with a homogeneous variable of degree 1 (i.e. only one variable). A linear equation can include several variables.

Here,

If c represents the number of cars and v represents the number of vans,

5c+8v=234

c=2v

10v+8v=234

v=234/18

v=13 vans

c=26 cars

The number of cars for field trip is 26 and number of vans is 13.

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Darnell is underwater 20 feet below sea level. His friend is on a hill 32 feet above sea level. How far apart are the two friends?

Answers

Answer:

52 feet apart

Step-by-step explanation:

20 + 32 = 52

Answer:

52 feet

Step-by-step explanation:

20 + 32 = 52

A cylinder has a base radius of 2 meters and a height of 2 meters what is its volume in cubic maters to the nearest tenths place

Answers

After answering the presented question, we can conclude that As a result, the cylinder's volume is about 25.1 cubic metres to the nearest tenths place.

what is cylinder?

A cylinder is a three-dimensional geometric shape made up of two parallel congruent circular bases and a curving surface connecting the two bases. The bases of a cylinder are always perpendicular to its axis, which is an imaginary straight line passing through the centre of both bases. The volume of a cylinder is equal to the product of its base area and height. A cylinder's volume is computed as V = r2h, where "V" represents the volume, "r" represents the radius of the base, and "h" represents the height of the cylinder.

The volume of a cylinder is calculated as V = r2h, where "r" is the radius of the base and "h" is the height.:

V = π

$(2^2)(2)$

V = 8π

Using a calculator and estimating π to 3.14, we get:

25.12 cubic metres V

When we round to the nearest tenths place, we get:

25.1 cubic metres (V).

As a result, the cylinder's volume is about 25.1 cubic metres to the nearest tenths place.

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Bat ampanget ko???????????

Answers

Answer:

Tanungin mo si face para malaman mo

What is the value of x in the equation 5(3x + 4) = 23?
43 over 15
three over 15
15 over 43
2 over 5

Answers

Answer:

B) three over 15

Step-by-step explanation:

Solve the equation:

5(3x + 4) = 2315x + 20 = 2315x = 23 - 2015x = 3x = 3/15

Correct choice is B

three over 15

Answer:

solution given:

5 (3x+4)=23

opening bracket

15x+20=23

15x=23-20

x=3/15

The scatter plot shows the number of households, in millions, that have cable television over eight consecutive years.

image of a scatter plot with x axis labeled Time in Years and y axis labeled Number of Households, with points at 1 comma 3 and 8 tenths, 2 comma 5 and 8 tenths, 3 comma 6 and 2 tenths, 4 comma 7 and 5 tenths, 5 comma 7 and 2 tenths, 6 comma 8 and 3 tenths, 7 comma 9 and 3 tenths, and 8 comma 8 and 5 tenths.

Predict the number of households with cable television in year 12.

8.9 million
9.7 million
12.1 million
14.5 million

Answers

Answer:

Plot the points on the graphing calculator. Then generate a linear regression model.

f(x) = .6714x + 4.0536

Substituting 12 for x:

f(12) = about 12.1 million

make q the subject of 4p=5q-2

Answers

Answer:

5q = 4p + 2

Step-by-step explanation:

4p = 5q - 2

4p + 2 = 5q - 2 + 2

5q = 4p + 2

Can someone please help me with math.

Answers

Answer:

Step-by-step explanation:

What was the temperature at sea level at 20,000 ft

Answers

Answer:

constant temperature is -56.5°C (-69.7°F), and this is also the lowest assumed temperature in respect to ISA.

if p(a) = 0.62, p(b) = 0.47, and p(a è b) = 0.88, then p(a ç b) =
a. 0.2914
b. 1.9700
c. 0.6700
d. 0.2100

Answers

Answer: a

Step-by-step explanation:a

Find X

Please help me answer this question and explain to me how you got that answer.Thank you.

Answers

Based on the two-tangent theorem, the value of x is calculated in the circle as: x = 2.

How to Find x Using the Two-Tangent Theorem?

Based on the two-tangent theorem, if two tangents are drawn from a circle to meet each other at any point outside the circle, then the lengths of the tangents are congruent to each other, that is, they are equal.

Therefore, we would create the following equation based on the two-tangent theorem:

20x + 6 = 10x + 26

Combine like terms:

20x - 10x = -6 + 26

10x = 20

Divide both sides by 10:

10x/10 = 20/10

x = 2

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pls help 20points if help.How would the growing season of plants in Florida differ from the growing season of plants in New York?

a.The growing seasons of both states are similar. This is because the climates of both states are similar.

b.The growing seasons of both states are the same. This is because the climate has very little effect on growing seasons.

c.The growing season in New York is much longer than the growing season in Florida. New York's seasons are warmer than seasons in Florida.

d.The growing season in Florida is much longer than the growing season in New York. Florida’s seasons are warmer than seasons in New York.

Answers

Answer:

D.The growing season in Florida is much longer than the growing season in New York. Florida’s seasons are warmer than seasons in New York.

Step-by-step explanation:

help me solve this question: 2/3(5q+2+6r)

Answers

Answer:

mommy harder yes pls ugHhH

There’s more options but what’s the answer for now

Answers

Answer:-

$\sqrt[3]{125x^{12}y^6}\\\\\\ \to \sqrt[3]{(5x^{4}y^{2})^{^{3}}}\\\\\\ \to 5x^4y^2$

What is the value of x-9 = -5

Answers

Answer:

$\boxed {x = 4}$

Step-by-step explanation:

Solve for the value of $x$:

$x - 9 = -5$

-Add both sides by $9$:

$x - 9 + 9 = -5 + 9$

$\boxed {x = 4}$

So, the value of $x$ is $4$.

which of the following functions are solutions of the differential equation y′′−9y′ 18y=0? a. y(x)=e6x b. y(x)=e−x c. y(x)=e3x d. y(x)=0 e. y(x)=6x f. y(x)=3x g. y(x)=ex

Answers

Only one of the following functions is a solution of the differential equation y′′−9y′+18y=0.

The second-order homogeneous linear differential equation is given as:y'' - 9y' + 18y = 0This differential equation is a linear homogeneous equation. We will have two roots of the characteristic equation: r1 = 3, r2 = 6So, the general solution to the differential equation is given as:y = c1e3x + c2e6xwhere c1 and c2 are arbitrary constants.a. y(x) = e6x is a solution because it is a part of the general solution of the differential equation.y(x) = e−x, y(x) = 0, y(x) = 6x, y(x) = 3x, y(x) = ex are not solutions because they don't satisfy the differential equation. Hence, the correct options are:a. y(x) = e6xTherefore, only one of the following functions is a solution of the differential equation y′′−9y′+18y=0.

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Use the order of operations to evaluate the numerical expression.

Answers

Answer:

I'm sorry but I need some numbers to solve. we just learned this in class so I'm new though

Step-by-step explanation:

Needs numbers

6/15=3/x.

I'm going insane

Answers

Answer:

x = 7.5

Step-by-step explanation:

$ \frac{6}{15} = \frac{3}{x} $

cross multiply expressions

6x = 45 divide both sides by 6

x = 7.5

Paco decides to invest $600 into a bank account that gives him a 3.4% interest rate compounded quarterly. He wants to know how long it will take for him to save up to $1,000.

Answers

Answer:

It would take 15 years

Step-by-step explanation:

The length of time it takes to reach $1,000 savings can be determined from the future value formula given below:

FV=PV*(1+r/4)^n*4

FV is the target savings of $1,000

PV is the amount invested which is $600

r is the rate of interest of 3.4%

n is the unknown

1000=600*(1+3.4%/4)^4n

1000=600*(1+0.0085 )^4n

1000=600*(1.0085)^4n

1000/600=1.0085^4n

1.666666667 =1.0085^4n

take log of both sides

ln 1.666666667 =4n ln 1.0085

4n=ln 1.666666667/ln 1.0085

4n=0.510825624 /0.008464078

4n=60.35218768

n=60.35218768 /4

n= 15.09

I need help with this please help me

Answers

Answer:

The graph is shown below.

Step-by-step explanation:

The given piecewise function is

$f(x)=\begin{cases}-x & \text{ if } x<0 \\ x^2 & \text{ if } x\geq 0 \end{cases}$

From the given piecewise function, it is clear that

$f(x)=-x$ for x<0.

Put different values of x in f(x) where x<0 and find the values of f(x) to make a table of values as shown below.

Table of values:

x f(x)

-3 3

-2 2

-1 1

Now, plot the points (-4,4),(-3,3),(-2,2),(-1,1) and join them by straight line. There is an open circle at x=0 because 0 is not included in this function.

From the given piecewise function, it is clear that

$f(x)=x^2$ for $x\geq 0$.

Table of values:

x f(x)

0 0

1 1

2 4

Plot the points (0,0), (1,1), (2,4) and join them by a curve. There is closed circle at 0 because 0 is included in this function.

The graph is shown below.

Solve for b. Round your answer to the nearest tenth.

A right triangle labeled A B C and A C B is a right angle. Segment A B is 15, segment B C is labeled a, segment A C is labeled b, and angle A B C is 26 degrees.

b = _(blank)_

Type your numerical answer below.

Answers

Answer: 6.57 ==> rounded of to 10

Step-by-step explanation:

sin B = sin 26 = 0.438

sin x = opp/hyp

0.438 = b / 15

b = 15 x 0.438 = 6.57

nearest 10th is 10

Last week, the price of apples at a grocery
store was $1.60 per pound. This week,
apples at the same grocery store are on sale
at a 10% discount. What is the total price
of 4% pounds of apples this week at the
grocery store?
A. $4.77
B. $6.48
C. $6.75
D. $6.93

Answers

The total price of 4 pounds of apples this week at the grocery store is $5.76

What is the total price after discount?

Last week price of apples per pound = $1.60Discount = 10%Quantity of apples = 4 pounds

Total price of apples this week = Last week price of apples per pound × Quantity of apples

= $1.60 × 4

= $6.4

Actual price paid = Total price of apples this week - (Discount × Total price of apples this week)

= $6.4 - (10% × $6.4)

= 6.4 - (0.1 × 6.4)

= 6.4 - 0.64

= $5.76

In conclusion, 4 pounds of apples cost $5.76 this week at the grocery store.

Answers

Answer:

Step-by-step explanation:

And let the missing number be x

We can solve using Algebra

11x - 10 = 34

11x = 34 + 10

11x = 44

x = 44/11

x = 4

For verification:

4(11) - 10 = 34

44 - 10 = 34

34 = 34

Hence, Proved

What is the marginal rate of substitution (MRS) for u(x)=5x
2

?

Answers

The marginal rate of substitution (MRS) for u(x) = 5x^2 is 10x.

The marginal rate of substitution (MRS) represents the rate at which a consumer is willing to trade off one good for another while keeping their utility constant. In the given utility function u(x) = 5x², where x represents the quantity of a particular good, we can determine the MRS by taking the derivative of the utility function with respect to x.

Taking the derivative of u(x) = 5x² with respect to x, we get du/dx = 10x. This derivative represents the MRS. The coefficient 10 indicates that for each unit increase in x, the MRS increases by a factor of 10.

The MRS of 10x implies that the consumer is willing to give up 10 units of the other good in exchange for an additional unit of x while maintaining the same level of utility. This indicates a constant rate of substitution between the two goods.

In summary, the marginal rate of substitution (MRS) for the utility function u(x) = 5x² is 10x, indicating the consumer's willingness to trade off the other good for x at a constant rate of substitution.

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The line plot displays the number of roses purchased per day at a grocery store.

A horizontal line starting at 1 with tick marks every one unit up to 10. The line is labeled Number of Rose Bouquets, and the graph is titled Roses Purchased Per Day. There is one dot above 1 and 10. There are two dots above 6, 7, and 9. There are three dots above 8.

Which of the following is the best measure of center for the data, and what is its value?

The mean is the best measure of center, and it equals 8.
The median is the best measure of center, and it equals 7.3.
The mean is the best measure of center, and it equals 7.3.
The median is the best measure of center, and it equals 8.

Answers

The answer choice that is the best measure of center for the data, and what is its value is D. The median is the best measure of center, and it equals 8.

How to calculate this?

A measure of center gives an idea of the central tendency of the data.

The median is a better choice here because it is not affected by extreme values, unlike the mean.

From the dots plotted on the line graph, it is evident that there are more data points around 8 than any other value. Hence, the median of the data would be closest to 8.