Determine whether the series is convergent or divergent by expressing sn as a telescoping sum.
[infinity]
Σ 8/n^2-1
n=3

Answers

Answer 1

Answer:

The sum converges at: [tex]\frac{10}{3}[/tex]

Step-by-step explanation:

Given

[tex]\sum\limits^{\infty}_{n =2} \frac{8}{n^2 - 1}[/tex]

Express the denominator as difference of two squares

[tex]\sum\limits^{\infty}_{n =2} \frac{8}{(n - 1)(n+1)}[/tex]

Express 8 as 4 * 2

[tex]\sum\limits^{\infty}_{n =2} \frac{4 * 2}{(n - 1)(n+1)}[/tex]

Rewrite as:

[tex]4 * \sum\limits^{\infty}_{n =2} \frac{2}{(n - 1)(n+1)}[/tex]

Express 2 as 1 + 1 + 0

[tex]4 * \sum\limits^{\infty}_{n =2} \frac{1+1+0}{(n - 1)(n+1)}[/tex]

Express 0 as n - n

[tex]4 * \sum\limits^{\infty}_{n =2} \frac{1+1+n - n}{(n - 1)(n+1)}[/tex]

Rewrite as:

[tex]4 * \sum\limits^{\infty}_{n =2} \frac{(n + 1)-(n - 1)}{(n - 1)(n+1)}[/tex]

Split

[tex]4 * \sum\limits^{\infty}_{n =2} \frac{(n + 1)}{(n - 1)(n+1)}-\frac{(n - 1)}{(n - 1)(n+1)}[/tex]

Cancel out like terms

[tex]4 * \sum\limits^{\infty}_{n =2} \frac{1}{(n - 1)}-\frac{1}{(n+1)}[/tex]

In the above statement, we have:

[tex]a_3 + a_5 = 4[(\frac{1}{2} - \frac{1}{4}) + (\frac{1}{4} - \frac{1}{6})][/tex]

[tex]a_3 + a_5 = 4[(\frac{1}{2} - \frac{1}{6})][/tex]

Add [tex]a_7[/tex]

[tex]a_3 + a_5 + a_7= 4[(\frac{1}{2} - \frac{1}{6}) + (\frac{1}{7 - 1} - \frac{1}{7+1})][/tex]

[tex]a_3 + a_5 + a_7= 4[(\frac{1}{2} - \frac{1}{6}) + (\frac{1}{6} - \frac{1}{8})][/tex]

[tex]a_3 + a_5 + a_7= 4[(\frac{1}{2} - \frac{1}{8})][/tex]

Notice that the pattern follows:

[tex]a_3 + a_5 + a_7 + ...... + a_{k}= 4[(\frac{1}{2} - \frac{1}{k+1})][/tex]

The above represent the odd sums (say S1)

For the even sums, we have:

[tex]4 * \sum\limits^{\infty}_{n =2} \frac{1}{(n - 1)}-\frac{1}{(n+1)}[/tex]

In the above statement, we have:

[tex]a_4 + a_6 = 4[(\frac{1}{3} - \frac{1}{5}) + (\frac{1}{5} - \frac{1}{7})][/tex]

[tex]a_4 + a_6 = 4[(\frac{1}{3} - \frac{1}{7})][/tex]

Add [tex]a_8[/tex] to both sides

[tex]a_4 + a_6 +a_8 = 4[(\frac{1}{3} - \frac{1}{7}) + \frac{1}{7} - \frac{1}{9}][/tex]

[tex]a_4 + a_6 +a_8 = 4[\frac{1}{3} - \frac{1}{9}][/tex]

Notice that the pattern follows:

[tex]a_4 + a_6 + a_8 + ...... + a_{k}= 4[(\frac{1}{3} - \frac{1}{k+1})][/tex]

The above represent the even sums (say S2)

The total sum (S) is:

[tex]S = S_1 + S_2[/tex]

[tex]S =4[(\frac{1}{2} - \frac{1}{k+1})] + 4[(\frac{1}{3} - \frac{1}{k+1})][/tex]

Remove all k terms

[tex]S =4[(\frac{1}{2}] + 4[(\frac{1}{3}][/tex]

Open bracket

[tex]S =\frac{4}{2} + \frac{4}{3}[/tex]

[tex]S =\frac{12 + 8}{6}[/tex]

[tex]S =\frac{20}{6}[/tex]

[tex]S =\frac{10}{3}[/tex]

The sum converges at: [tex]\frac{10}{3}[/tex]


Related Questions

What is the value of X

A. x=20

B. x=15

C. x=10

D. x=5

Answers

The answer is A) x=20.

This is because (x+5) + (7x-5) + x = 180

Ethan is 1.6 metres tall, Uzma is 154 cm tall.
Work out how much taller Ethan is than Uzma.

Answers

Answer:

1.6 metres is 160 centimetres

Ethan is 6 centimetres taller than Uzma

lim x->1+( sin(1-x)-(e^(x-1))+1)/ lnx

Answers

We're given the one-sided limit,

[tex]\displaystyle\lim_{x\to1^+}\frac{\sin(1-x)-e^{x-1}+1}{\ln(x)}[/tex]

Evaluating the limand directly at x = 1 gives the indeterminate from

(sin(1 - 1) - exp(1 - 1) + 1) / ln(1) = 0/0

so we can potentially solve the limit by applying L'Hopital's rule. Doing so gives

[tex]\displaystyle\lim_{x\to1^+}\frac{\sin(1-x)-e^{x-1}+1}{\ln(x)}=\lim_{x\to1^+}\frac{-\cos(1-x)-e^{x-1}}{\frac1x}=\frac{-\cos(0)-e^0}{\frac11}=\boxed{-2}[/tex]

write an equation of the line below ​

Answers

I think the equation is y= -x - 4

Explanation:

The two given coordinates are (0,-4) and (4,-7).

Use those to find the slope which is -1

Then use the slope intercept form and substitute the answers where they belong.

y - y1 = m(x - x1)

y - (-4) = -1 (x - 0)

y = -x - 4

Jake wants to surprise his parents with a small anniversary party at their favorite restaurant.

It will cost $35.75 per person for dinner, including tip and tax.

His budget for the party is $600.

What is the maximum number of people Jake can have at the party without exceeding his budget?

Answers

Answer:

16 people

Step-by-step explanation:

35.75 times 16 is 572, times 17 is 607.75, he only has 600 and he cant have a fraction of a person, so he can have 16 people

35.75x=600
x/35.75=600/35.75
x=16.78

He can invite 16 people without exceeding his budget.

There are 3 liters of orange juice at a school party. 10 students want to drink all of the orange juice, and they all want to get exactly the same amount. How much orange juice can each get? (enter your answer as a fraction or mixed number)

Answers

Answer:

3 1/3 each

Step-by-step explanation:

10/3=3 1/3

0.3 liters each student

3 liters = 3.000 ml so 3.000(ml) : 10(students) =300 ml which is 0.3 liters

Meg makes a dot plot for the data 9, 9, 4, 5, 5, 3,
4,5, 3, 8, 8, 5. Where does a gap occur?

Answers

The gap consists of the values 6 and 7.

Check out the dot plot below to see what I mean. We have one cluster on the left from 3 to 5. Then another cluster on the right from 8 to 9.

Answer:

The gap is present around 6 & 7

Step-by-step explanation:

Since there is no evidence of dots above 6 & 7, there must have been a freeze in data around then, called a gap.

4
)
5 + 15x
42x + 4
A) -2
C) -10
B) 3
D) -8

Answers

Answer:

D

Step-by-step explanation:

D

if a student is selected at random, find the probability that the student is 14 year old​

Answers

Answer:

If the student is selected at random the answer is if you put 14 years apart of the student selected at random means the answer are not apart.

Step-by-step explanation:

14 years selected the problem is that 14-9 is more then 9 and u add that up to get the right equation for the 14 years apart

Simplify the given expression and enter in numerical form.6+5

Answers

Answer:

6+5=11

Step-by-step explanation:

6 is added to 5

i am sorry if i am wrong

4/8 =?/2 please answer​

Answers

Answer:

? = 1

Step-by-step explanation:

4/8 = ?/2

Change the ? into a variable so it's easier to calculate:

Variable x = ?

4/8 = x/2

Cross multiply:

4 × 2 = 8 × x

8 = 8x

Divide both sides by 8 to isolate the variable:

1 = x

Check your work:

4/8 = 1/2

4 × 2 = 8 × 1

8 = 8

Correct!

1/2 Numerator and denominator are both divided by 4

the product of 2 numbers, p and q, decreased by 3 times their sum. (as an algebraic expression)

Answers

Answer:

3 - (p*q) im not sure lol

Step-by-step explanation:

Find the slope in each line

Answers

can you put the YX just because it’s too blurry.

Keith is looking at a cliff. He determines that the angle of elevation to the top is 70° from where he is at. 70m away from Keith, Alan estimates the angle between the base of the cliff, himself, and Keith to be 29° while Keith estimates the angle between the base of the cliff, himself, and Alan to be 48°. What is the height, h, of the cliff to the nearest tenth of a metre?
1) 78.8m
2) 83.9m
3) 89.0m
4) 95.7m

This question completely baffled me on my homework, I still haven't done it because I don't even know how to approach it (Since it's a 3D problem)
Please help me out
Also, this homework is due in an hour so I'm starting to sweat

Answers

9514 1404 393

Answer:

  4) 95.7 m

Step-by-step explanation:

The angle at the cliff base between Alan and Keith is ...

  B = 180° -29° -48° = 103°

The law of sines will tell you the distance KB from Keith to the cliff base is

  KB/sin(A) = KA/sin(B)

  KB = KA×sin(A)/sin(B) = (70 m)sin(29°)/sin(103°) ≈ 34.82935 m

The height of the cliff is found from the tangent relation ...

  Tan = Opposite/Adjacent

  tan(70°) = height/(34.83 m)

  height = (34.83 m)tan(70°) = 95.693 m

The height of the cliff is about 95.7 m.

_____

Additional comment

These problems can usually be resolved into two 2-D problems. Here, we can work out the distance we need from Keith to the cliff by considering only the ground plan of the site. Then we can work out the height of the cliff only considering the vertical plane containing Keith and the base of the cliff.

The attachment shows the ground plan triangle KAB.


Two and three fifths plus one and three fifths

Answers

Answer:

2 3 + 1 3 = 3/5 + 3/5 = 1 1/5 + 2+1 = 3 + 1 !/5 = 4 1/5

5 5

Step-by-step explanation: The Slashes mean The number bellow

Please make this answer Brainlist...

Which is the graph of an even monomial function?

Answers

Answer:

The graph of an even function is symmetric about the y-axis. The graph of an odd function is symmetric about the x-axis. It is possible that the use of these two words originated with the observation that the graph of a polynomial function in which all variables are to an even power is symmetric about the y -axis.

please help me!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Answers

9514 1404 393

Answer:

  the correct answer is marked

Step-by-step explanation:

The circle with center (h, k) and radius r has equation ...

  (x -h)² +(y-k)² = r²

For the given center, that is ...

  (x -1)² +(y +1)² = r²

The value of r² can be found by using the given point for x and y, and seeing what value of r² makes the equation true.

  (5 -1)² +(7 +1)² = 4² +8² = 16 +64 = 80

So, r² = 80 makes the equation true, and that equation is ...

  (x -1)² +(y +1)² = 80

Tamara uses a coupon to save 20% off her purchase at the bookstore. The original cost of Tamara's purchase is d dollars.
Which expressions represent the cost, in dollars, of Tamara's purchase after using the coupon? Choose all that apply.

Answers

Answer:

The equation would be d - (d×0.2)

using data from 2010 and projected to 2020, the population of the United Kingdom (y, in millions) can be approximated by the equation 10.0y-4.55x=581. where x is the number of years after 2000 what is projected population in 2026?

Answers

Answer:

The projected population in 2026 is of 699.3 million.

Step-by-step explanation:

Population:

The population, in x years after 2000, is given by the follwing equation:

[tex]y - 4.55x = 581[/tex]

That is:

[tex]y(x) = 4.55x + 581[/tex]

What is projected population in 2026?

2026 - 2000 = 26, so this is y(26).

[tex]y(26) = 4.55(26) + 581 = 699.3[/tex]

The projected population in 2026 is of 699.3 million.

Two functions are shown in the table below f(x) equals -X squared plus 4X +12 And G(x)
equals X +2

Answers

Answer:

[tex]x = -2; x = 5[/tex]

Step-by-step explanation:

Given

[tex]f(x) = -x^2 + 4x + 12[/tex]

[tex]g(x) = x +2[/tex]

Required

For what value of x is: [tex]f(x) = g(x)[/tex]

[tex]f(x) = g(x)[/tex] implies that

[tex]-x^2 + 4x + 12 = x + 2[/tex]

Collect like terms

[tex]-x^2 + 4x -x + 12 -2 = 0[/tex]

[tex]-x^2 + 3x + 10 = 0[/tex]

Expand

[tex]-x^2 + 5x-2x + 10 = 0[/tex]

Factorize

[tex]-x(x - 5)-2(x - 5) = 0[/tex]

Factor out x - 5

[tex](-x - 2)(x - 5) = 0[/tex]

Solve for x

[tex]-x - 2 = 0; x - 5 = 0[/tex]

So:

[tex]x = -2; x = 5[/tex]

My rule is: y = 1/3x + 11/15

Find y, if x=1

Find y, if x=6

Answers

Mark brainliest please

Answer: if x=1 , then y= 48/45
If x=6 then y= 42/15

Explaination

If x=1

Replacing value of x in the equation

y= 1/3*1 + 11/15

y= 48/45


If x= 6

Replacing value of x

y= 1/3*x + 11/15

= 42/15

ACTIVITY 1 Directions: Interpret these circle graphs

A. The pie graph below shows what a normal human body is made of.


Questions:

1. As shown in the graph,what composes the human body?

2. Which makes up the greatest part of the human body? the least?

Answers

Step-by-step explanation:

1) it is composed of 20% of fat, 17% of bone, 50% of muscle and 13% others(minerals)

2) muscle makes up the greatest part and the minerals makes up the least.

hope it heals u.

make me brain list Pls.

PLEASE HELP ASAP PLSSSSS !!!

Answers

Answer:

133 answer ---------------------------------------- helpful answer

QUESTION 14 · 1 POINT
Translate the English phrase into an algebraic expression: 6x less than 81xsquared

Answers

Answer:

Step-by-step explanation:

81x² - 6x

Answer:

81x² - 6x

Step-by-step explanation:

81x² - 6x

76. In the diagram below, lines land mare
parallel. Both are intersected by
transversal t.
What is the value of x?

Answers

4x = 2x + 30 Alternate angles in parallel lines are equal
2x = 30
x = 15

Please help me please please I really need help please please

A moving company charges $30 plus $0.15 per mile to rent a moving van. Another company charges $15 plus $0.20 per mile to rent the same van. For how many miles will the cost be the same for the two companies? Write and solve an equation.

Answers

30 + .15x = 20 + .20x
10 = .05x
200= x
200 miles

According to a government study, among adults in the 25- to 34-year age group, the mean amount spent per year on reading and entertainment is $2,020. Assume that the distribution of the amounts spent follows the normal distribution with a standard deviation of $456. Use Appendix B.3. (Round your z-score computation to 2 decimal places and final answers to 2 decimal places.) What percent of the adults spend more than $2,600 per year on reading and entertainment

Answers

Answer:

z = x-u/s

z = 2600 - 2020/456

z = 1.27

 = .897

1-.898 = .102

z > 1.27 = 10.2%

The percentage of adults spending more than $2,600 per year on reading and entertainment is 10.2%.

What is the z-score?

The standard score in statistics is the number of standard deviations that a raw score deviates from or exceeds the mean of the phenomenon being observed or assessed. Standard scores are positive for raw scores above the mean and negative for raw scores below the mean.

Given that according to a government study, among adults in the 25- to 34-year age group, the mean amount spent per year on reading and entertainment is $2,020. Assume that the distribution of the amounts spent follows the normal distribution with a standard deviation of $456.

The percentage will be calculated as,

z = x-u/s

z = 2600 - (2020/456)

z = 1.27

The value of z=1.27 in the table is 0.897. The percentage will be calculated as,

P = 1-.898 = .102

z > 1.27 = 10.2%

Therefore, the percentage of adults spending more than $2,600 per year on reading and entertainment is 10.2%.

To know more about z-score follow

https://brainly.com/question/25638875

#SPJ5

Choose all of the symbols that make the following sentence true.


4 ^0 ___ 5 ^-1


=

<







>

Answers

Answer:

≠ and >

Step-by-step explanation:

_________________

Three rectangular prisms each have a height of 1 cm.
. Prism A has a base that is 1 cm by 11 cm
* Prism B has a base that is 2 cm by 7 cm SA +V
Prism C has a base that is 3 cm by 5 cm.
1. Find the surface area and volume of each prism. Use the dot paper to draw the prisms, if needed.

Answers

Answer:

A: SA = 46 cm^2; V = 11 cm^3

B: SA = 46 cm^2; V = 14 cm^3

C: SA = 46 cm^2; V = 15 cm^3

Step-by-step explanation:

SA = 2B + PH = 2LW + PH

where SA = total surface area,

B = area of a base

P = perimeter of the base

H = height of the prism

L = length of the base

W = width of the base

V = LWH

Prism A:

SA = 2(11 cm)(1 cm) + 2(11 cm + 1 cm)(1 cm)

SA = 46 cm^2

V = (11 cm)(1 cm)(1 cm) = 11 cm^3

Prism B:

SA = 2(7 cm)(2 cm) + 2(7 cm + 2 cm)(1 cm)

SA = 46 cm^2

V = (7 cm)(2 cm)(1 cm) = 14 cm^3

Prism C:

SA = 2(5 cm)(3 cm) + 2(5 cm + 3 cm)(1 cm)

SA = 46 cm^2

V = (5 cm)(3 cm)(1 cm) = 15 cm^3

Triangle D E F is shown. Angle E F D is a right angle. The length of E F is 24 and the length of D F is 7.
Which trigonometric ratios are correct for triangle DEF? Select three options.


sin(D) = StartFraction 24 Over 25 EndFraction
cos(E) = StartFraction 7 Over 25 EndFraction
tan(D) = StartFraction 24 Over 7 EndFraction
sin(E) = StartFraction 7 Over 25 EndFraction
tan(D) = StartFraction 7 Over 24 EndFraction

Answers

Answer:

Sin(E) = 7/25

Sin(D) = 24/25

Tan(D)= 24/7

Step-by-step explanation:

Sin=opp/hyp

Tan=opp/adj

Cos=adj/hyp

The trigonometric ratios that are correct for triangle DEF are sin(E) = 7/25, sin(D) = 24/25 and tan(D)= 24/7

How to determine the trigonometric ratios?

The given parameters are:

EF = 24

DF = 7

Start by calculating the length DE using:

DE²= EF² + DF²

So, we have:

DE²= 24² + 7²

Take the square root of both sides

DE= 25

The sine of the angles is calculated using:

sin(Ф) = opp/hyp

So, we have:

sin(E) = 7/25

sin(D) = 24/25

The tangent of the angles is calculated using:

tan(Ф) = opp/adj

So, we have:

tan(D)= 24/7

Hence, the trigonometric ratios that are correct for triangle DEF are sin(E) = 7/25, sin(D) = 24/25 and tan(D)= 24/7

Read more about trigonometric ratios at:

https://brainly.com/question/11967894

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