Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that
Rn(x) → 0.] Find the associated radius of convergence R.
f(x) = 8(1 − x)^−2
show step by step including finding the derivatives.

Answers

Answer 1

Recall that for |x| < 1, we have

[tex]\displaystyle \frac1{1-x} = \sum_{n=0}^\infty x^n[/tex]

Differentiating both sides gives

[tex]\displaystyle \frac1{(1-x)^2} = \sum_{n=0}^\infty nx^{n-1} = \sum_{n=0}^\infty (n+1)x^n[/tex]

and multiplying both sides by 8 gives the series for f(x) :

[tex]f(x)=\displaystyle \frac8{(1-x)^2} = \boxed{8\sum_{n=0}^\infty (n+1)x^n}[/tex]

and this converges over the same interval, |x| < 1, so that the radius of convergence is 1.


Related Questions

The line l with equation x - 2y + 2 = 0 crosses the y-axis at the point P. The line
m with equation 3x + y - 15 = 0 crosses the y-axis at the point Q and intersects
l at the point R. Find the area of triangle PQR.​

Answers

Answer:

Area of ΔPQR is 28 units²

Step-by-step explanation:

-P is the point with coordinates ( 0, y-intercept for line x-2y+2 =0)

-rearrange the equation in the point-slope form y=mx+b  to find the y coordinate of the point P( 0, b)

x-2y+2 = 0, subtract x and 2 from both sides

-2y = -x-2, divide by -2 both sides

y= (1/2)x +1 so b=1 and P (0, 1)

-Q  is the point with coordinates ( 0, y-intercept for line 3x+y -15 =0)

-rearrange the equation in the point-slope form y=mx+b  to find the y coordinate of the point Q( 0, b)

3x +y -15 =0, subtract 3x and add 15 to both sides

y= -3x +15 so b=15 and Q(0,15)

-R is the intersection of the two lines so is the solution of the system of equations y= (1/2)x +1 and y= -3x +15

(1/2)x +1 =  -3x +15, add 3x and subtract1

(1/2) x+3x = 15-1, combine like terms

(7/2)x = 14 , multiply both sides by 2

7x = 28, divide both sides by 7

x= 4

y= (1/2)x +1 = (4/2) +1 =3 so R(4,3)

- the area of ΔPQR is (base *height)/2

base= 15-1= 14

height = 4

A= (14*4)/2 = 14*2 = 28

Which of the following is the point and slope of the equation y + 14 = 7(x - 18)?

Answers

Answer:

y = 7x - 140

The slope is 7

The y-intersept is -140

= (7, -140)

Step-by-step explanation:

y + 14 = 7(x - 18)

y + 14 = 7x - 126

y =7x - 126 - 14

y = 7x - 140

Britany wants to read a book. In her room, she has 5 mysteries, 15 historical fictions, 12 modern fantasies, and 7 blographies.
How many different choices are available?


pleaseee

Answers

Answer:

39

Step-by-step explanation:

5 + 15 + 12 + 7 = 39

Answer:

39

Step-by-step explanation:

5 + 15 = 20, 12 + 7 = 19, 20 + 19 = 39.

The director of special events for Sun City believed that the amount of money spent on fireworks displays on the 4th of July was predictive of attendance at the Fall Festival held in October. She gathered the following data to test her suspicion.

4th of July ($000) Fall Festival (000)
10.6  8.8
 8.5  6.4
12.5 10.8
 9.0 10.2
 5.5   6.0
12 11.1
 8.0   7.5
 7.5   8.4
 9.0   9.5
10   9.8
 7.5   6.6
10 10.1
 6.0   6.1
12 11.3
10.5   8.8

Required:
Determine the regression equation. Is the amount spent on fireworks related to attendance at the Fall Festival? Conduct a hypothesis test to determine if there is a problem with autocorrelation.

Answers

Answer:

The complete question and its solution in the attached file please find it.

Step-by-step explanation:

Identify which of the following is not equivalent to 234.

Question 1 options:

A)




B)

23−−√4

C)

(214)12

D)

214 x 212

Answers

Answer:

A since A is left blank its A

Which expression is equal to
(3x – 4)(2x – 5)?

Answers

Answer:

6x^2-23x+20

Step-by-step explanation:

i think the expanded form of that equation is equal to it.

(3x-4)(2x-5)

3x(2x-5)-4(2x-5)

6x^2-15x-8x+20

6x^2-23x+20

I hope this helps and sorry if it's wrong

6x^2-23x+20 sorry if it’s wrong I’m not sure

Do number 7 plz finding QR​

Answers

Answer:

QR = sqrt(147)

Step-by-step explanation:

Since this is a right triangle, we can use the Pythagorean theorem

a^2 + b^2 = c^2

7^2 + QR^2 = 14^2

49 + QR^2 = 196

QR^2 = 196-49

QR^2 = 147

Taking the square root of each side

QR = sqrt(147)

Which graph represents y = |xl?
A
B
C
D

Answers

Answer:

B

Step-by-step explanation:

The equation represented in the question is the parent absolute value question. If you know the different parent functions, then the answer is obvious because absolute value equations always form a V. However, if you do not do this then you can create a table and plugin values. Plugin numbers like 0, -1, and 1 for X and solve for Y. Finally, graph these points and see what graph best fits. If needed you can also plug in more points.

Answer:

B.

Step-by-step explanation:

I got it correct on the warm up

Using the Time Series Data for the US Nuclear Electric Power Production, calculate the exponential smoothing forecasts for 2005 through 2010 using alpha equals 0.2. Find the forecast error for each time period. If there is n

Answers

Answer:

????????????

I don't know

If there is a die that has 12 sides, that are numbered 1 to 12, what is the probability that she will roll either a 3 or a 9

Answers

Answer:

2/12 = 1/6

Step-by-step explanation:

To find the probability of something with an equal chance of each outcome, we can apply the formula (number of favorable outcomes)/(number of total outcomes). Because there is an equal chance for each side of the die to be landed on, we can apply this.

On a 12 sided die, there are 12 sides. Two of those sides are 3 and 9. Therefore, there are two favorable outcomes (3 and 9). There are 12 sides to choose from, so there are 12 total outcomes, making the probability 2/12 = 1/6

Find the area of athletic field if it's length is 120cm and its width is 28cm .A. 397.6cm B. 3360cm C. 296 cm D. 4592cm E. 3356cm​

Answers

Answer:

B 3360

Step-by-step explanation:

Area of Rectangle = Length X Width

120 X 28

= 3360 cm

Answered by Gauthmath

Select the correct answer.
Which is the minimum or maximum value of the given function?
dndnsn

Answers

Answer:

See explanation

Step-by-step explanation:

The question is incomplete, as the function is not given. So, I will make an assumption.

A quadratic function is represented as:

[tex]f(x) = ax^2 + bx + c[/tex]

If [tex]a > 0[/tex], then the function has a minimum x value

E.g. [tex]f(x) = 4x^2 - 5x + 8[/tex] ------ [tex]4 > 0[/tex]

Else, then the function has a maximum x value

E.g. [tex]f(x)= -4x^2 -5x + 8[/tex] ---- [tex]-4 < 0[/tex]

The maximum or minimum x value is calculated using:

[tex]x = -\frac{b}{2a}[/tex]

For instance, the maximum of [tex]f(x)= -4x^2 -5x + 8[/tex] is:

[tex]x = -\frac{-5}{2*-4}[/tex]

[tex]x = -\frac{5}{8}[/tex]

So, the maximum of the function is:

[tex]f(x)= -4x^2 -5x + 8[/tex]

[tex]f(-\frac{5}{8}) = -4 * (-\frac{5}{8})^2 - 5 *(-\frac{5}{8}) +8[/tex]

[tex]f(-\frac{5}{8}) = 9.5625[/tex]

Find the medien: 16,12,10,15,7,9,16

Answers

Answer:

12

Step-by-step explanation:

arrange the numbers in ascending order and cross out from either side till you have a middle line

FINAL ANSWER:

12

Step-by-step explanation:

Median is the middle number in the data set.

so first of ... we need to arrange the group of numbers from lower to greater.

16, 12, 10, 15, 7, 9, 16 ⇒ 7, 9, 10, 12, 15, 16, 16

Now that we have arranged the numbers from least to greatest all we need to do is to find the middle number of the data set (data set? they are the group of numbers)

Ok, so what you want to do here is to just count the numbers until you get to the middle number of the data set...

7, 9, 10, 12, 15, 16, 16

the median in the given data set is 12.

I hope this helps you!!! Let me know if my answer is incorrect or not...

HAVE A GREAT DAY AND GOD BLESS YOU ;)!!!

What is the value of the expression (2x + y) (2x - y) when x = 4 and y = -5?

Answers

Answer:

39

Step-by-step explanation:

1. (2(4)-5)(2(4)+5)

2.(3)(13)

3.39

Answer:

Step-by-step explanation:

This is a difference of squares question. You should 64 = 25 = 39 Let's see if that happens.

Difference of squares

(2x - y) ( 2x + y) = 4x^2 - y^2

4(4)^2 - (5)^2

64 - 25 = 39

Now do the question exactly as it is written.

(2*4 - -5)(2*4 + -5)

(8 +5)(8 - 5)

3 * 13

39

They really do give the same answer.

Nadia needs 3/4 cup of orange juice for a punch recipe. She will double the recipe to make
punch for a party. Which statement is true?

Answers

Answer:

she will be using more orange juice

Answer:

she will be using more orange juice

Find the slope of the line #67

Answers

Vas happenin
Hope your day is going well
The slope form is y2-y1/x2-x1
So I’m going to use (1,1) (-1,4)
Y2= 4
Y1=1
X2= -1
X1= 1
Then you plug in those equations 4-1/ -1 -1
3/-2 that’s the fraction form
The decimal form is -1.5 this is the slope
Hope this helps *smiles*

I need help with this ​

Answers

Answer:

D.

Step-by-step explanation:

According to converse of the Pythagorean Theorem, that if the square of the third side (longest side) of a triangle equals the sum of the other two shorter sides, therefore, the triangle formed must be a right triangle.

From the side lengths given, the following satisfies the Pythagorean triple:

5² + 12² = 13².

Therefore, the procedure to use to confirm the converse of the Pythagorean Theorem would be to draw the two shortest side, 5 cm and 12 cm, so that a right angle will be between them. The side measuring 13 cm should therefore fit in to form a right triangle.

Simplify: 3.5 x 10^-2 + 2.3 x 10^-2

Answers

Given:

The given expression is:

[tex]3.5\times 10^{-2}+2.3\times 10^{-2}[/tex]

To find:

The simplified form of the given expression.

Solution:

We have,

[tex]3.5\times 10^{-2}+2.3\times 10^{-2}[/tex]

It can be written as:

[tex]=(3.5+2.3)\times 10^{-2}[/tex]

[tex]=5.8\times 10^{-2}[/tex]

Therefore, the simplified form of the given expression is [tex]5.8\times 10^{-2}[/tex].

Solve this equation for x. Round your answer to the nearest hundredth. 0.77 = log x​

Answers

Answer:

x ≈ 5.89

General Formulas and Concepts:

Pre-Algebra

Equality Properties

Algebra II

Exponential to Logarithmic:                                                                             [tex]\displaystyle b^m=x \rightarrow log_bx=m[/tex]

Step-by-step explanation:

Step 1: Define

Identify

0.77 = log(x)

Step 2: Solve for x

[Equality Property] Raise both sides to the 10th power:                                [tex]\displaysytle 10^{0.77} = 10^{logx}[/tex]Simplify:                                                                                                             [tex]\displaysytle x = 10^{0.77}[/tex]Evaluate:                                                                                                           [tex]\displaysytle x = 5.88844[/tex]

There are three separate, equal-size boxes, and inside each box there are four separate small boxes, and inside each of the small boxes there are three even smaller boxes. how many boxes are there all together?

Answers

Answer:

There are 51 boxes all together.

Step-by-step explanation:

Since there are three separate, equal-size boxes, and inside each box there are four separate small boxes, and inside each of the small boxes there are three even smaller boxes, to determine how many boxes are there all together, the following calculation must be done:

3 + 3 x 4 + 3 x 4 x 3 = X

3 + 12 + 36 = X

51 = X

Therefore, there are 51 boxes all together.

A trough has ends shaped like isosceles triangles, with width 2 m and height 5 m, and the trough is 18 m long. Water is being pumped into the trough at a rate of 8 m3/min. At what rate (in m/min) does the height of the water change when the water is 2 m deep

Answers

9514 1404 393

Answer:

   5/9 m/min

Step-by-step explanation:

The depth of the water is 2/5 of the depth of the trough, so the width of the surface will be 2/5 of the width of the trough:

  2/5 × 2 m = 4/5 m

Then the surface area of the water is ...

  A = LW = (18 m)(4/5 m) = 14.4 m²

The rate of change of height multiplied by the area gives the rate of change of volume:

  8 m³/min = (14.4 m²)(h')

  h' = (8 m³/min)/(14.4 m²) = 5/9 m/min

The value of 33 + 42 = ___.

Numerical Answers Expected!

Answer for Blank 1:

Answers

Answer:

3^3 + 4^2 = 43

Step-by-step explanation:

Please write the numbers like this 3^3 + 4^2 next time

3^3 + 4^2 =

Solve for 3^3

3^3 = 3 • 3 • 3 = 27

Solve for 4^2

4^2 = 4 • 4 = 16

Add 27 and 16

27 + 16 = 43

[tex] \sf Q) \: {3}^{3} + {4}^{2} = {?}[/tex]

[tex] \sf \to \: {3}^{3} + {4}^{2} [/tex]

[tex] \sf \to \: 27 + 16= 43 [/tex]

Thus, the value is 43.

find the constant of variation k for the direct variation. 4x = -y

Answers

Answer:

k = - 4

Step-by-step explanation:

The equation representing direct variation is

y = kx ← k is the constant of variation

Given

4x = - y ( multiply both sides by - 1 )

- 4x = y , that is

y = - 4x ← in standard form

with k = - 4

The theoretical mean of a distribution is also known as its ______________.

Answers

Answer:

skewness

Step-by-step explanation:

Average.

The average of a set of observations is the most important and useful measure of statistics and is a position measure, as it shows the positions of the numbers to which it refers. The average value is involved in several types of statistics and is examined in almost all statistical distributions. It is generally defined as the sum of the observations by their number. That is, it is the mathematical operation of finding the "mean distance" between two or more numbers.

Learn more about averages in https://brainly.com/question/22390452

Michael drove 210 miles in 3 1/2. Jordan drove 330 miles in 6 hours. Which is an accurate comparison of the rates at which the two people drove?

Answers

Michael = 210 / 3.5 = 60 miles per hour

Jordan = 330/ 6 =55 miles per hour

Jordan drove 5 miles per hour slower than michael

construct an angle that bisect 120°​

Answers

Answer:

just make a 120 angle and divide it by 2, 60.

20 slips of paper are put into a bag numbered from 1 to 20. One slip is randomly selected from the bag. We are interested in selecting even numbers. What is the probability of selecting an even number from the bag?

Answers

Answer:

2/5

Step-by-step explanation:

Answer:

Step-by-step explanation:

There are 10 even numbers from 1 to 20

2 4 6 8 10 12 14 16 18 20

There 20 possible choices.

P(Even) = 10 /20 = 1/2

Perimeter (numerical)

Answers

Answer:

270 m

Step-by-step explanation:

Add up all the sides

P = 19 +18.8+18.8 +18.8+18.8+40.8+19+40.8+18.8+18.8+18.8+18.8

P = 270 m

Simplify 7a – 11b + 4ab – 6a + 5b.

Answers

7a-6a-11b+5b+4ab
=a-6b+4ab
Final answer

The wholesale price of 6 oz plastic bottles is 6 cents how many plastic bottles can be purchased for $98.41

Answers

Answer:

1640

Step-by-step explanation:

Take the total amount and divide by the amount for one

Make sure to write 6 cent in dollar form (.06)

98.41 / .06

1640.1666

Round down since we need to buy whole bottles

1640

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