find the interval of convergence of the power series. (be sure to include a check for convergence at the endpoints of the interval.) (7x)^n/(10n)!

Answers

Answer 1

The interval of convergence of the power series is (-10/7, 10/7).

What does Interval of convergence tell you?

A group of x-values on which a power series converges is known as the interval of convergence. In order to create a convergent series, you can plug in this interval of x-values.

The interval of convergence of a power series is the set of values of x for which the series converges. To find the interval of convergence of the given power series, we can use the ratio test.

The ratio test states that if the series [tex]∑ a_nx^n[/tex] has a positive radius of convergence R, then the series converges for |x| < R and diverges for |x| > R. The radius of convergence is given by the formula

                     [tex]R = 1/limsup|a_n+1/a_n|.[/tex]

In the given power series, the coefficient [tex]a_n[/tex] is given by  [tex]a_n = (7x)^n/(10n)!.[/tex]

We can rewrite this as [tex]a_n = x^n(7/10)^n/n!.[/tex]

Taking the limit of the ratio [tex]a_n+1/a_n[/tex] as n goes to infinity, we get:

             [tex]limsup|a_n+1/a_n| \\\\= limsup|x^(n+1)(7/10)^(n+1)/(n+1)!/(x^n(7/10)^n/n!)|\\\\= limsup|x(7/10)/(n+1)|\\\\= |x(7/10)|[/tex]

Therefore, the radius of convergence R is [tex]1/|x(7/10)|[/tex].

This means that the interval of convergence is  [tex]|x| < 1/|7x/10| = 10/7.[/tex]

We can check the endpoints of the interval to see if the series converges at these points. At x = 10/7, the series becomes [tex](7(10/7))^n/(10n)! = 1^n/(10n)!.[/tex]

This series is a geometric series with first term 1 and common ratio 1/10, and it converges because [tex]|1/10| < 1.[/tex]

At x = -10/7, the series becomes[tex](7(-10/7))^n/(10n)! = (-1)^n/(10n)![/tex]. This series is an alternating series with first term 1 and common ratio (-1/10), and it also converges because [tex]|-1/10| < 1.[/tex]

Therefore, The interval of convergence of the power series is (-10/7, 10/7).

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Related Questions

What if the value of <5

Answers

Answer:

any number from -infinity to 4 is the value less then 5

If 43x-2=4 x+1, then x

Answers

Answer:

Step-by-step explanation:

Picture has the answer, not a bad picture I promise.

Step-by-step explanation:

43x-4x=1+2

39x/39=3/39

x=0.076

Solve for q: 2q- 12 = -40​

Answers

Answer:

q = -14

Step-by-step explanation:

2q - 12 = -40

Add 12  to both sides

2q - 12 + 12 = - 40 + 12

2q  = - 28

Divide both sides by 2

2q/2 = -28/2

q = - 14

which graph correspond to the following-2x-3y​

Answers

Answer:

Hi! The answer to your question is in the picture below

Step-by-step explanation:

☆*: .。..。.:*☆☆*: .。..。.:*☆☆*: .。..。.:*☆☆*: .。..。.:*☆

☆Brainliest is greatly appreciated!☆

Hope this helps!!

- Brooklynn Deka

Which expressions are equivalent to the one below? Check all that apply. 5^3 • 5^x
A. 5^3 + x
B. 125•5^3
C. (5 • x)^3
D. 25^3x
E. 53^3x
F. 53^3-x​

Answers

Hi there, I'll help you out :)

The expression that is equivalent to 5^3 • 5^x is A. If you solve the expression given in the question, you get 5^3 + x, which is exactly the same as the first choice.

- B gives you 15625

- C gives you 125x^3

- D gives you 15625^x

- E gives you 148,877^x

- F can't really be solved

Because none of the other answer choices match up, that leaves A as your only answer.

Hope this helps!

The expressions which are equivalent to the one given below are 5³⁺ˣ and 125 (5ˣ).

What are Expressions?

Expressions are mathematical statements which consist of two or more terms and terms are connected to each other using mathematical operators like addition, multiplication, subtraction and so on.

Given expression is,

5³ . 5ˣ

We have the product rule of exponents that,

xᵃ . xᵇ = xᵃ⁺ᵇ

Using the same here,

5³ . 5ˣ = 5³⁺ˣ ------(1)

Now the value of 5³ = 5 × 5 × 5 = 125

So, 5³ . 5ˣ = 125 (5ˣ) --------(2)

Hence the equivalent expressions are 5³⁺ˣ and 125 (5ˣ).

Learn more about Expressions here :

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Find an equation in slope-intercept form of the line that has slope 2 and passes through point A(-3. - 1)

Answers

Answer:

Step-by-step explanation:

m = 2 ; x1 = -3 ; y1 = -1

Slope- point form: y - y1 = m(x -x1)

y - [-1] = 2(x - [-3])

y + 1 = 2(x + 3)

y + 1 = 2x + 3*2

y + 1 =2x + 6

   y = 2x + 6 - 1

  y = 2x + 5  This is slope intercept form : y = mx + b

please anwser this thank you!!​

Answers

Step-by-step explanation:

v>29 or 29,infinity sign i used an app called and this is what it showed

<3<3<3<3<3<3<3<3<3<3<<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3

Answer:

v >_ 29

<3<3<3<3<3<3<3<3<3<3<<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3<3

What’s the volume of 9 cm 6 cm 4 cm

Answers

Answer:

bro tell me the shape

Step-by-step explanation:

The mean height of women in a country (ages 20-29) is 63.9 inches. A random sample of 65 women in this age group is selected. What is the probability that the mean height for the sample is greater than 64 inches? Assume σ = 2.96.

Answers

Answer: The mean height of women in a country ages 20 29

The mean height of women in a country (ages 20-29) is 63.7 inches.

please do this thanks

Answers

Answer:

225 in²

Step-by-step explanation:

First you have to find the ∛3,375 since it is a cube you know that all the dimensions are the same so when 15 comes out all you have to do is multiply the height by the length to get the area of one of its sides to get the answer.

need help with math its a struggle

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Answer:

(1,4)

(0,0)

Step-by-step explanation:

smarts

Answer:

1,4

0,0

Step-by-step explanation:

whatever is in the purple is valid and the closed line is inclusive so anything on it counts but the dotted line is exclusive

Luna's dining room is 19 feet wide and 22 feet long. Luna wants to install wooden trim
around the top of the room. The trim costs $0.06 per foot. How much will it cost Luna to buy
enough trim?

Answers

9514 1404 393

Answer:

  $4.92

Step-by-step explanation:

The cost is found by multiplying the number of feet by the cost per foot. The number of feet is the perimeter of the dining room, the sum of the lengths of all the sides.

There are two sides of the rectangular room that are 19 feet, and two sides that are 22 feet, so the total length of trim is ...

  P = 2(L +W) = 2(22 ft + 19 ft) = 2(41 ft) = 82 ft

The cost is then ...

  cost = ($0.06/ft)(82 ft) = $4.92

It will cost Luna $4.92 to buy enough trim.

which expression is equivalent to 6^4x6^3

Answers

Answer:

C - (6x6x6x6)(6x6x6)

Step-by-step explanation:

David rode his bike a total of 214 miles from his house to the store. First he rode 34 mile from his house to his school. Then he rode 14 mile from his school to the library. Finally he rode from the library to the store. How many miles did David ride between the library and the store?

Answers

Answer:

166 miles

Step-by-step explanation:

Add the 34 miles he rode from home to school and the 14 miles he rode from school to the library.

34 + 14 = 48.

Subtract 48 from 214 to get 166

214 - 48 = 166

David rode 166 miles between the library and the store.

Hope this helps! Pls mark brainliest

13/4 + c = 3/4 what’s c,?

Answers

Answer:

c = -2.5

Step-by-step explanation:

13/4 + c =3/4

2.5 + c = 0

c=-2.5

A child can choose one stuffed animal. She can choose from a dog or a bear. Each comes in brown, gray or white. You can also choose green eyes or blue eyes. How many possible stuffed animal choices does the child have?

Answers

Step-by-step explanation:

[tex]2 \times 3 = 6 \\ 6 \times 2 = 12[/tex]

8 The circle graph below shows the number of votes each candidate received in a recent student
council election.
Student Council Election
Votes Received
Jacob
60
Elias
25
Jade
40
What percentage of the votes did Jacob receive?
F 36%
G 44%
H 48%
J 60%

Answers

Answer:

60%

Step-by-step explanation:

4. What is the volume of the triangular pyramid?
10 in
6 in
8 in
Will Mark brainliest

Answers

Answer:

V=6240 in^3

Step-by-step explanation:

I am not sure what the base is but here is my guess:

V=13Ah

V=13(48)(10)

V=6240 in^3

With double-digit annual percentage increases in the cost of health insurance, more and more workers are likely to lack health insurance coverage (USA Today, January 23, 2004). The following sample data provide a comparison of workers with and without health insurance coverage for small, medium, and large companies. For the purposes of this study, small companies are companies that have fewer than 100 employees. Medium companies have 100 to 999 employees, and large companies have 1000 or more employees. Sample data are reported for 50 employees of small companies, 75 employees of medium companies, and 100 employees of large companies.
Health Insurance
Size of Company Yes No Total
Small 33 17 50
Medium 68 7 75
Large 88 12 100
Conduct a test of independence to determine whether employee health insurance coverage is independent of the size of the company. Use = .05.
1. Compute the value of the 2 test statistic (to 2 decimals).
2. The p value is :_____.
a. less than .005
b. between .005 and .01
c. between .01 and .025
d. between .025 and .05
e. between .05 and .10
f. greater than .10
3. What is your conclusion?
a. health insurance coverage is not independent of the size of the company
b. Cannot reject the assumption that health insurance coverage and size of the company are independent
4. The USA Today article indicated employees of small companies are more likely to lack health insurance coverage. Calculate the percentages of employees without health insurance based on company size (to the nearest whole number).
Small %
Medium %
Large %

Answers

Answer:

1.  χ² =  15.3902

2. The p value is :_____.a. less than .005

3.  Conclusion

a. health insurance coverage is not independent of the size of the company

4. The percentages of employees

Small %=   33/50= 0.66

Medium %= 68/75 = 0.91

Large %= 88/100= 0.88

Step-by-step explanation:

1) We set up our null and alternative hypothesis as

H0: the employee health insurance coverage is independent of the size of the company

against the claim

Ha: the employee health insurance coverage is not independent of the size of the company

2) the significance level alpha is set at 0.05

3) the test statistic under H0 is

χ²= ∑ (O - E)²/ E where O is the observed and E is the expected frequency

which has an approximate chi square distribution with ( 3-1) (2-1)=  2 d.f

4) Computations:

Under H0 , the observed frequencies are :

Observed       Expected E        (O-E)         (O-E)²             (O-E)²/E

33                   42                      -9               81                    1.928  

68                  63                         5              25                  0.3968

88                  84                         4               16                  0.1904

17                    8                           9              81                  10.125

7                     12                          -5             21                    1.75

12                    16                         -4              16                     1              

                                                                                          15.3902    

Expected Values are calculated using the formula :

Row Total * Column Total / sample size

E1= (33+17) (33+ 68+88)/ 50+75+100= 42

E4= (33+17) (17+ 7+ 12)/ 50+75+100=8

E2= (68+7) (33+ 68+88)/ 50+75+100= 63

E5= (68+7)  (17+ 7+ 12)/ 50+75+100= 12

E3= (88+12) (33+ 68+88)/ 50+75+100= 84

E6= (88+12)  (17+ 7+ 12)/ 50+75+100= 16

5) The critical region is χ² ≥ χ² (0.05)2 = 5.99

6) Conclusion:

The calculated χ² =  15.3902    falls in the critical region χ² ≥  5.99  so we reject the null hypothesis that the employee health insurance coverage is NOT independent of the size of the company.

2. The p value is :_____.

a. less than .005

The p-value is .000385.

3.  Conclusion

a. health insurance coverage is not independent of the size of the company

4. The percentages of employees

Small %=   33/50= 0.66

Medium %= 68/75 = 0.91

Large %= 88/100= 0.88

Use the box plots for 3 and 4. What are the two medians. Pls help me

Answers

Answer:

C

Step-by-step explanation:

Answer:

C. 40 and 50

Step-by-step explanation:

The median marks the mid-point of the data and is shown by the line that divides the box into two parts (sometimes known as the second quartile). Half the scores are greater than or equal to this value and half are less.

Which of the following inequalities is correct​

Answers

B is correct

Converted and written correctly, your options are
A: 12.5 pounds = 200 ounces
B: 490 ounces > 480 ounces (= 30 pounds)
C: 6 tons = 12,000 pounds > 10,000 pounds
D: 8,000 pounds = 4 tons

PLEASE HELP !! ILL GIVE BRAINLIEST *EXTRA 40 POINTS* DONT SKIP :(( .!

Answers

Answer:

angle 8 and 4

Step-by-step explanation:

it is shown by the picture I had just provided you

Answer:

A

Explination:

Hope this helps ^-^

What is the missing length?
A. 15
B. 30
C. 8
D. 7

Answers

Answer:

red and blue are not a Selection!!!!!

Please help!!!!!!!!!

Answers

Answer:

-1 is the remainder

hope it helps

A survey was conducted to investigate whether there is an association between a person's age and their thoughts on what a state senate should do about state parks. Participants selected from "leave state parks the way they are," "increase funding to state parks," or "completely overhaul state parks," and their age was categorized as under 30 years old, between 30 and 50 years old, and over 50 years old. The hypothesis test statistic was calculated to be 15.01. Which of the following is closest to the p- value of the test?
A. 0.0047
B. 0.0103
C. 0.0202
D. 09897
E. 0.9953

Answers

Answer:

E. 0.9953

Step-by-step explanation:

The computation of the p value is shown below:

According to the question

The participant choosen from 3 different state parks, r = 3

And there are 3 different age group, s = 3

So, degrees of freedom is

= (r-1) × (s-1)

= 2 × 2

= 4

And, the Test statistic is 15.01

Now

p-value = P(Chisq < 15.01 )

= 0.9953

In the chisquare table see the degree of freedom i.e. 4

Answer:

0.0047

Step-by-step explanation:

This is what college board says

The value 0.0047 is the probability that χ2 is greater than 15.01 in an χ2 distribution with (3−1)(3−1)=4 degrees of freedom.

Long division259560÷25​

Answers

Answer:

21

Step-by-step explanation:

10382.4 is the right answer please give me brainly

Select all numbers that have an absolute value of 4.

Answers

wheres the numbers at?

Pls help 10 ptsssssssssssss
NO LINKS

Answers

Answer:

the answer is 16

Step-by-step explanation:

8/2=4

4*4=16

the area of the triangle is 16

I need some help! Will give brainliest!!!

Answers

Answer:

Positive 7 on the line pointing up (y axis) is where you start (where your first dot goes when graphing). Then down two and right one. (Your second dot will end up on (5,1))

Step-by-step explanation:

Amiyah has two coupons from Macy's Department Store, one for $20 off
and another for 20% off. She wants to purchase a shirt for $64. Her
friend Emma told her to use the 20% off coupon because it is the best
deal and her other friend Jermaine told her to use the $20 off coupon
because it is the best deal. Which friend is correct and why?

Answers

Answer:Jermaine

Step-by-step explanation:

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