Let f be defined by the function f(x) = 1/(x^2+9)
(a) Evaluate the improper integral [tex]\int\limits^{∞}_3 {f(x)} \, dx[/tex] or show that the integral diverges
(b) Determine whether the series ∑n=3∞ f(n) converges or diverges State the conditions of the test used for determining convergence or divergence
(c) Determine whether the series ∑n=1∞(−1)n(en⋅f(n))=∑n=1∞(−1)n(n2+9)en converges absolutely, converges conditionally, or diverges (image put below)
![Let F Be Defined By The Function F(x) = 1/(x^2+9)(a) Evaluate The Improper Integral [tex]\int\limits^{}_3](https://brainimage.s3.us-east-005.backblazeb2.com/contents/attachments/yzZjpXHkLAE7AjHcTpwkE8ZmnkZwgsU7.png)
Answers
(a)
[tex]\displaystyle\int_3^\infty \frac{\mathrm dx}{x^2+9}=\lim_{b\to\infty}\int_{x=3}^{x=b}\frac{\mathrm dx}{x^2+9}[/tex]
Substitute x = 3 tan(t ) and dx = 3 sec²(t ) dt :
[tex]\displaystyle\lim_{b\to\infty}\int_{t=\arctan(1)}^{t=\arctan\left(\frac b3\right)}\frac{3\sec^2(t)}{(3\tan(t))^2+9}\,\mathrm dt=\frac13\lim_{b\to\infty}\int_{t=\arctan(1)}^{t=\arctan\left(\frac b3\right)}\mathrm dt[/tex]
[tex]=\displaystyle \frac13 \lim_{b\to\infty}\left(\arctan\left(\frac b3\right)-\arctan(1)\right)=\boxed{\dfrac\pi{12}}[/tex]
(b) The series
[tex]\displaystyle \sum_{n=3}^\infty \frac1{n^2+9}[/tex]
converges by comparison to the convergent p-series,
[tex]\displaystyle\sum_{n=3}^\infty\frac1{n^2}[/tex]
(c) The series
[tex]\displaystyle \sum_{n=1}^\infty \frac{(-1)^n (n^2+9)}{e^n}[/tex]
converges absolutely, since
[tex]\displaystyle \sum_{n=1}^\infty \left|\frac{(-1)^n (n^2+9)}{e^n}\right|=\sum_{n=1}^\infty \frac{n^2+9}{e^n} < \sum_{n=1}^\infty \frac{n^2}{e^n} < \sum_{n=1}^\infty \frac1{e^n}=\frac1{e-1}[/tex]
That is, ∑ (-1)ⁿ (n ² + 9)/eⁿ converges absolutely because ∑ |(-1)ⁿ (n ² + 9)/eⁿ| = ∑ (n ² + 9)/eⁿ in turn converges by comparison to a geometric series.
Related Questions
A pound of chocolate costs 8 dollars. Yoko buys p pounds. Write an equation to represent the total cost c that Yoko pays.
Answers
Answer:
8×p≈c
Step-by-step explanation:
Because if each is 8 dollars you would multpy the number (Or Letter) by how many you want or have.
in certain mill the cost of c in thousands of dollars of producing x tons of steel is given by c(x)= 0.03x +3.4. the revenue R in thousands of dollars by selling x tons is given R(x)= 0.5x. Find x the number of tons that must be produced to break even
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Answer:
well you see if u wanrt to
a
If angle A in the diagram equals 45 degrees and angle B is 4x + 5 degrees, determine the value of x.
A. 10 Degrees
B. 25/2 Degrees
C. 40 Degrees
D. 160 Degrees
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ANSWER FAST!! All of the following angle pairs are ALWAYS congruent when a transversal passes through 2 parallel lines EXCEPT:
Adjacent angles
Alternate Interior Angles
Alternate Exterior Angles
Corresponding Angles
Answers
Answer:
Adjacent angles
Step-by-step explanation:
Find the simplified form of the expression. Give your answer in scientific notation.
(8 x 107) (7 x 104)
Answers
880 x 700
= 616000
=6.16 x 10^5
Using the same survey methodology: random sample of US residents between the ages of 18 to 75 who registered on the Crest website as a product, Crest wants to divide the research population into heterogeneous groups based on age and gender and then draw a random sample from each of the different groups. Based on this information, what type of sampling method would Crest use to conduct this survey
Answers
Answer:
Stratified sampling
Step-by-step explanation:
Samples may be classified as:
Convenient: Sample drawn from a conveniently available pool.
Random: Basically, put all the options into a hat and drawn some of them.
Systematic: Every kth element is taken. For example, you want to survey something on the street, you interview every 5th person, for example.
Cluster: Divides population into groups, called clusters, and each element in the cluster is surveyed.
Stratified: Also divides the population into groups. However, then only some elements of the group are surveyed.
In this question:
Population divided into groups, and random samples are drawn from the groups. This is a mark of stratified sampling.
I need help with this!
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Concept Recall #2: (6.RP.A.1)
There are 14 girls and 15 boys in a sixth grade math class.
Write this relationship as a ratio.
Answers
Answer:
b
Step-by-step explanation:
Answer:
B is the correct answer
Step-by-step explanation:
Hope this helps
somebody PLEASE help, 2x - 8 = 12
what does x =??
Answers
What is the factored form of the polynomial?
x2 – 16x + 48
(x − 4)(x − 12)
(x − 6)(x − 8)
(x + 4)(x + 12)
(x + 6)(x + 8)
Answers
Answer:
(x − 4)(x − 12)
Step-by-step explanation:
The factored form of the polynomial x² - 16x + 48 is (x - 4)(x - 12).
Option A is the correct answer.
Polynomial is an equation written with terms of the form kx^n.
where k and n are positive integers.
There are quadratic polynomials and cubic polynomials.
Example:
2x³ + 4x² + 4x + 9 is a cubic polynomial.
4x² + 7x + 8 is a quadratic polynomial.
We have,
Now we can rewrite the middle term -16x as -4x - 12x and group the terms as follows:
x² - 16x + 48 = x² - 4x - 12x + 48
= (x² - 4x) + (-12x + 48)
= x(x - 4) - 12(x - 4)
= (x - 4)(x - 12)
Therefore,
The factored form of the polynomial x² - 16x + 48 is (x - 4)(x - 12).
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giving brainliest for correct answer
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Answer:
Option A 64/27 cm³
Step-by-step explanation:
Side of each cube=1/3 cm
Volume of a cube=side³=(1/3)³=1/27 cm³
Total no. of cubes in the prism=4*4*4=64 cubes
Volume of prism=Volume of 1 cube*Total no. of cubes
=(1/27)*64
=64/27 cm³
Hope this helps
Help me out pls and thank you very much !!!!!!!
Answers
Answer:
8
Step-by-step explanation:
Since this is a rectangle, the opposite sides are congruent.
So the lengths of DG and EF are equal
3x + 5 = 29
3x = 29 - 5
3x = 24
x = 24/3
x = 8
Answer please..........
Answers
Answer:
Rasheed has sent 66 massages , Hassan has sent 22 , Shima has sent 29
Expand and simplify
4(x + 2y + 3(x + y)
Answers
Answer is in the file below
tinyurl.com/wpazsebu
Answer:
The answer is 16x+20y for the above expression
2. APPLES: Mr. Kelly's homeroom went picking. The line plot show's the number of pounds of apples that were collected.
a. Describe the shape of the distribution. Identify any clusters, gaps, peaks, or outliers.
b. Describe the center and spread of the distribution. Justify your response based on the shape of the distribution.
(NOT A MULTIPLE CHOICE)
Answers
Answer:
a) shape of distribution: symmetric/unimodal, clusters: none, gaps: none, peaks: 20, outliers: none.
b) The center and spread of distribution is based solely on the median of the data because it is based on a data point that is actually included in the population or sample. The median in this data is 20 because it is between the first and third quartile.
Q13. Mavi Jeans has two retail outlets: Dallas and Phoenix. The Dallas store does 65 percent of the total sales in a year. Further analysis indicates that if a sale is made in Dallas, there is 0.5 probability this is a sale of belts, while if a sale is made at the Phoenix store this probability is 0.3. If Mavi Jeans knows that a belt has been sold, what is the probability this sale has occurred in Dallas
Answers
Answer:
0.7558 = 75.58% probability this sale has occurred in Dallas
Step-by-step explanation:
Conditional Probability
We use the conditional probability formula to solve this question. It is
[tex]P(B|A) = \frac{P(A \cap B)}{P(A)}[/tex]
In which
P(B|A) is the probability of event B happening, given that A happened.
[tex]P(A \cap B)[/tex] is the probability of both A and B happening.
P(A) is the probability of A happening.
In this question:
Event A: Belt sold
Event B: Sold in Dallas.
Probability of belt being sold:
50% of 65%(made in Dallas)
30% of 100 - 65 = 35%(made in Phoenix). So
[tex]P(A) = 0.5*0.65 + 0.3*0.35 = 0.43[/tex]
Sold in Dallas:
50% of 65%. So
[tex]P(A \cap B) = 0.5*0.65 = 0.325[/tex]
What is the probability this sale has occurred in Dallas?
[tex]P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.325}{0.43} = 0.7558[/tex]
0.7558 = 75.58% probability this sale has occurred in Dallas
Hoshi walks 10 meters in 3 seconds. She walks the same at this rate for 50 seconds. How far does she walk?
Answers
x meters ----> 50 sec
50(10) = 3x
500 = 3x
devide both in 3
x = 166.66 m
a parabola can be drawn given a focus (4,-11) and a directrix of y=7. write the equation of the parabola in any form
Answers
Answer: The vertex will have the same x coordinate as the focus
and be at the midpoint between focus and directrix. The
midpoint of (4, 4) and (4, -2) is: ((4+4)/2, (4-2)/2) = (4, 1)
help with math thanks soo much will give brainlyist!!!!!
Answers
Answer:
b=3
Step-by-step explanation:
21 divded by 7=3 since area is Lenght times Width
An article in Engineering Horizons (Spring 1990, p. 26) reported that 117 of 484 new engineering graduates were planning to continue studying for an advanced degree . Consider this as a random sample of the 1990 graduating class.
(A) Find a 90% confidence interval on the proportion of such graduates planning to continue their education.
(B) Find a 95% confidence interval on the proportion of such graduates planning to continue their education.
(C) Compare your answers to parts (a) and (b) and explain why they are the same or different.
(D) Could you use either of these confidence intervals to determine whether the proportion is actually 0.25? Explain your answer. Hint: Use the normal approximation to the binomial.
Answers
Answer:
a) The 90% confidence interval on the proportion of such graduates planning to continue their education is (0.2097, 0.2737).
b) The 95% confidence interval on the proportion of such graduates planning to continue their education is (0.2036, 0.2798).
c) Due to higher value of z, a 95% confidence interval, as in part b, is wider than a 90% confidence interval, found in part a, and is the difference between these two intervals.
d) Yes, because 0.25 is inside both intervals.
Step-by-step explanation:
Question a:
In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence level of [tex]1-\alpha[/tex], we have the following confidence interval of proportions.
[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
In which
z is the zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex].
117 of 484 new engineering graduates were planning to continue studying for an advanced degree.
This means that [tex]n = 484, \pi = \frac{117}{484} = 0.2417[/tex]
90% confidence level
So [tex]\alpha = 0.1[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.1}{2} = 0.95[/tex], so [tex]Z = 1.645[/tex].
The lower limit of this interval is:
[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.2417 - 1.645\sqrt{\frac{0.2417*0.7583}{484}} = 0.2097[/tex]
The upper limit of this interval is:
[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.2417 + 1.645\sqrt{\frac{0.2417*0.7583}{484}} = 0.2737[/tex]
The 90% confidence interval on the proportion of such graduates planning to continue their education is (0.2097, 0.2737).
Question b:
95% confidence level
So [tex]\alpha = 0.05[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.05}{2} = 0.975[/tex], so [tex]Z = 1.96[/tex].
The lower limit of this interval is:
[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.2417 - 1.96\sqrt{\frac{0.2417*0.7583}{484}} = 0.2036[/tex]
The upper limit of this interval is:
[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.2417 + 1.96\sqrt{\frac{0.2417*0.7583}{484}} = 0.2798[/tex]
The 95% confidence interval on the proportion of such graduates planning to continue their education is (0.2036, 0.2798).
(C) Compare your answers to parts (a) and (b) and explain why they are the same or different.
The margin of error of a confidence interval of proportions is:
[tex]M = z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
This means that, due to higher value of z, a 95% confidence interval, as in part b, is wider than a 90% confidence interval, found in part a, and is the difference between these two intervals.
(D) Could you use either of these confidence intervals to determine whether the proportion is actually 0.25? Explain your answer. Hint: Use the normal approximation to the binomial.
Yes, because 0.25 is inside both intervals.
POSSIBLE
Name the relationship: complementary, supplementary, vertical, or adjacent.
1)
2)
b
b
A) supplementary
B) complementary
C) corresponding
D) alternate interior
A) complementary
B) vertical
C) alternate exterior
D) alternate interior
3)
4)
I
Answers
Write two fractions that have a sum of 1 and have different denominators.
Answers
Answer:
1/5+8/10=1
Step-by-step explanation:
1/5 = 2/10
Convince yourself that there are 3 similar triangles in the diagram. Choose the
similarity statement that represents the relationship among the 3 triangles.
Please only answer if your sure
Answers
Answer:
second option.
Step-by-step explanation:
What is the Area of this quadrilateral
Answers
C is the answer i just did this answer and get it correct
In the figure, m∠4=74° and m∠3=43° . Find m∠1 and m∠2 .
Answers
The measure of the angle m∠1 is 31 degree and the measure of the angle m∠2 is 106 degree.
What is an angle?When two lines or rays converge at the same point, the measurement between them is called a "Angle."
We have:
m∠4=74° and m∠3=43°
m∠4 + m∠2 = 180
74 + m∠2 = 180
m∠2 = 106 degree
m∠3 + m∠2 + m∠1 = 180 (sum of angles in triangle is 180 degree)
43 + 106 + m∠1 = 180
m∠1 = 31 degree
Thus, the measure of the angle m∠1 is 31 degree and the measure of the angle m∠2 is 106 degree.
Learn more about the angle here:
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Point 2 is 106°
Point 1 is 31°
Bit by bit clarification:
every triangle's total angle is 180
first you 180 - 74 = 106 for angle 2
angle 2 you had to use the angle two total then you add it with angle three like this 106 + 43 = 149 then you do 180 - 149 = 31 will give you your angle one
hope it help mark brainliset please
Will give brainliest if the answer is correct
Answers
Answer:
The answer to 5 is 3 and the answer to 6 is -2
Step-by-step explanation:
-8=2x-14
add 14 to each side
6=2x
divide by 2
3=x
----------------------------
11x+13=-9
subtract 13 from both sides
11x=-22
divide each side by 11
x=-2
Answer:
5.x=3
6.x=-2
Step-by-step explanation:
you're welcome
help plsssssssssssssssssssssssssssss
Answers
Answer:
c. x= 28°, y= 76°
Step-by-step explanation:
y= 180-104=76°
104= x+y (property of triangle, exterior angles is equal to the sum of opposite interior angles)
x=104-76= 28°
Johnny had 600 red apples and 400 blue oranges, then his friend gave him 20 red bananas. He eats all of the bananas, throws away 200 of the blue oranges because he couldn't finish them all, and he fed 100 of the red oranges to his pet rock. How many pumpkin pies does he have now?
Answer choices:
8,000,000
No, he has 6000 bananas
I think he just ordered some offline
He has 300 red apples
Answers
Answer:
4 im not sure
Step-by-step explanation:
(NO LINKS!!!!!!!!!!!!!!!!!!) 25 POINTS
Answers
Answer:
Definitely not "feels"!
Step-by-step explanation:
I would go with " Always" aka "A"
Answer:
I think C, never irrational
Step-by-step explanation:
I feel you, link scams are horrible.
If this answer helped, please give brainliest! It would help a lot if you gave it :D
Is 1.5399818181... a rational number?
Answers
Answer:
yes, cuz the rational numbers is said to be 0 or real numbers.