Confirm that the Integral Test can be applied to the series. Then use the Integral Test to determine the convergence or divergence of the series. Σ_(n=1)^[infinity] 2/(3n+5)
Answers
Σ_(n=1) [infinity]²/³ⁿ⁺⁵ is divergent.
A more detailed explanation of the answer.
To apply the Integral Test, we must first confirm that the function is positive, continuous, and decreasing. Let's consider the function f(n) = 2/(3n+5).
1. Positivity: Since n is positive and the denominator is also positive, the function f(n) is positive for all n in the given range.
2. Continuity: f(n) is a rational function, and there are no discontinuities in the given range.
3. Decreasing: The derivative of f(n) can be found as follows:
f'(n) = -6/(3n+5)^2
Since the denominator is always positive, f'(n) is negative, which means that f(n) is decreasing.
Since f(n) meets all three conditions, we can apply the Integral Test.
Now, let's evaluate the integral:
∫[1, ∞] 2/(3n+5) dn
To solve this integral, use the substitution method:
Let u = 3n+5
Then, du = 3 dn, so dn = du/3
Now, the integral becomes:
(2/3) ∫[1, ∞] 1/u du
The integral of 1/u is ln|u|. So,
(2/3) [ln|u|] evaluated from 1 to ∞
When the upper limit approaches infinity, the natural logarithm function (ln) also approaches infinity. Therefore, the integral is divergent.
Since the integral is divergent, by the Integral Test, the series is also divergent:
Σ_(n=1)²/³ⁿ⁺⁵ is divergent.
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Related Questions
A shipping container will be used to transport several 120-kilogram crates across the country by rail. The greatest weight that can be loaded into the container is 27500 kilograms. Other shipments weighing 12400 kilograms have already been loaded into the container. Which inequality can be used to determine x, the greatest number of
120-kilogram crates that can be loaded onto the shipping container?
Answers
Inequality can be used to determine x, the greatest number of 120-kilogram crates that can be loaded onto the shipping container is 8000 + 120C <= 27500
What is inequality?An inequality is a relationship that allows us to contrast two or more mathematical expressions.
Knowing that;
Each carton weighs 120 kg.
The container's maximum weight when loaded is 27500 kg.
Weight of the container as it is currently loaded: 8000 kg
Now that we have merged, we have;
Solution
because 8000 grams have already been loaded kilograms
into the container
each crate weighs 120 kilograms.
so 8000 + 120C <= 27500
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Using rectangles whose height is given by the value of the function at the midpoint of the rectangle's base (the midpoint rule), estimate the area under the graph of the following function, using two and then four rectangles. y=81-x^2 between x=-9 and x=9
Answers
The estimated area under the graph of the function y = \(81 - x^2\)using two rectangles is 1,094.5 square units, and using four rectangles is 513 square units.
To estimate the area under the graph of the function y = \(81 - x^2\)using the midpoint rule, we can divide the interval [-9, 9] into equal subintervals and approximate the area using rectangles.
First, let's consider using two rectangles. We divide the interval into two equal subintervals: [-9, 0] and [0, 9]. The midpoint of the first subinterval is x = -4.5, and the midpoint of the second subinterval is x = 4.5.
For each subinterval, the height of the rectangle is given by the value of the function at the midpoint. Plugging in the midpoint values, we have:
For the first subinterval [-9, 0]:
Height = 81 - (-4.5)^2 = 81 - 20.25 = 60.75
For the second subinterval [0, 9]:
Height = 81 - (4.5)^2 = 81 - 20.25 = 60.75
The width of each rectangle is (9 - (-9)) / 2 = 9.
Now, we can calculate the approximate area using the formula: Area ≈ height * width.
For two rectangles, the total area is:
Area ≈ (60.75 + 60.75) * 9 = 1,094.5 square units.
Next, let's consider using four rectangles. We divide the interval into four equal subintervals: [-9, -3], [-3, 3], [3, 9]. The midpoints of the subintervals are x = -6, x = 0, and x = 6.
For each subinterval, we calculate the height at the midpoint and multiply it by the width.
For the first subinterval [-9, -3]:
Height =\(81 - (-6)^2 = 81 - 36 = 45\)
Width = (-3 - (-9)) / 2 = 3
For the second subinterval [-3, 3]:
Height =\(81 - (0)^2 = 81\)
Width = (3 - (-3)) / 2 = 3
For the third subinterval [3, 9]:
Height = \(81 - (6)^2 = 81 - 36 = 45\)
Width = (9 - 3) / 2 = 3
Now, we can calculate the approximate area using the formula: Area ≈ (height1 + height2 + height3) * width.
Area ≈ (45 + 81 + 45) * 3 = 513 square units.
Thus, the estimated area under the graph of the function y = \(81 - x^2\)using two rectangles is 1,094.5 square units, and using four rectangles is 513 square units.
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(x - 10) + (3x -6) how do you solve this?
Answers
The expression (x - 10) + (3x - 6) is equivalent to the expression 4x - 16.
What is an equivalent expression?The equivalent is the expression that is in different forms but is equal to the same value.
The definition of simplicity is making something simpler to achieve or grasp while also making it a little less difficult.
The expression is given below.
⇒ (x - 10) + (3x - 6)
Simplify the expression, then we have
⇒ (x - 10) + (3x - 6)
⇒ x - 10 + 3x - 6
⇒ 4x - 16
The expression (x - 10) + (3x - 6) is equivalent to the expression 4x - 16.
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36+5 to the 3rd power as an expression
Answers
Answer:
the expression is (36+5)³
The probability of winning a certain game is 0. 5. If at least 70 percent of the games in a series of n games are won, the player wins a prize. If the possible choices for n are n=10, n=20, and n=100, which value of n should the player choose in order to maximize the probability of winning a prize?.
Answers
0.117 Or 11.7% chance of n should the player choose in order to maximize the probability of winning a prize.
What is the binomial theorem on probability?The probability of precisely x successes on n further trials in an experiment with two potential outcomes is referred to as the binomial probability (commonly called a binomial experiment).
When a procedure is performed a certain number of times (for example, in a group of patients), the result for each patient can either be a success or a failure, the binomial distribution model enables us to calculate the probability of witnessing a defined number of "successes."
The probability of winning a prize remains the same for. n = 10 or n=20 or n = 100; the probabilities are the same.
Given that,
p = 0.5, So, q = 1 - 0.5 = 0.5.
If 70% of 'n' games are won then 30% of n games must be lost.
Let n = 10.
Therefore, P ( x = 3) = \(^{10}C_3p^7q^3\)
or, P ( x = 3) = \(^{10}C_3(0.5)^7(0.5)^3\)
or, P (x = 3) = 120×0.0078×0.125.
or, P (x = 3) = 0.117 Or 11.7% chance.
Now, Changing the value won't have any effect as all the 'n's are multiple of 10 and we'll have same probability.
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5. in which number is the 3-1/100 the value of the 3 in 43,781? a. 345 b. 237, 895 c. 5,732
Answers
The number in which the value of 3 is 3-1/100 is 345. The correct option is a.
The given number is 43,781.
The value of the 3 in 43,781 is 3,000.
The value of 3-1/100 is 2.99.
We are supposed to find out in which number 2.99 is the value of the 3 in 43,781.
Given options are:
a. 345
b. 237,895
c. 5,732
We'll calculate the values of 3-1/100 in each option.
a. The value of 3-1/100 in 345 is 2.99.
So, the correct option is a.
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find the sum of the following 1.) 26+ (-30) + 13 =
Answers
Step-by-step explanation:
9 is the answer
26+ (-30) + 13 =
26-30 +13 =
-4 + 13 = 9
hope it's helpful to you
Consider a regular hexagon centered at (3,4). Describe a rotation that
will map this regular hexagon onto itself.
Answers
Answer:
36- degree rotation m8
Step-by-step explanation:
mark me brainliest :)
Discussion Topic You can identify sample spaces for compound events using organized lists, tables, and tree diagrams. Which of the three methods do you find easiest to use? Which method is the most helpful? Why? Use the Internet or another resource to find the definition of the Fuilidamental Counting Principle. What does this principle state? How can the principle be used to help you identify a sample space for a compound event? What are the limitations of using the Fundamental Counting Principle when determining the probability of an outcome? Support your answers with an example
Answers
I find organized lists to be the easiest method to use to identify sample spaces for compound events. This is because organized lists are the most straightforward way to list all of the possible outcomes of an event.
What is Fundamental Counting Principle?Tables and tree diagrams can be helpful as well, but they can be more difficult to create and interpret.
The Fundamental Counting Principle states that if there are n ways to do one thing, and m ways to do another thing, then there are n × m ways to do both things. This principle can be used to help identify a sample space for a compound event by multiplying the number of ways each event can occur. For example, if you are rolling a die and flipping a coin, there are 6 ways to roll the die and 2 ways to flip the coin. Therefore, there are 6 × 2 = 12 possible outcomes of the compound event.
The Fundamental Counting Principle is a useful tool for identifying sample spaces, but it does have some limitations. One limitation is that it only applies to events that are independent. Independent events are events where the outcome of one event does not affect the outcome of the other event. For example, the outcome of drawing a card from a deck does affect the outcome of drawing another card from the deck. In this case, the Fundamental Counting Principle cannot be used to determine the sample space.
Another limitation of the Fundamental Counting Principle is that it does not take into account the probability of each outcome. The probability of an outcome is the likelihood that the outcome will occur. For example, the probability of rolling a 6 on a die is 1/6. The probability of flipping a coin and getting heads is 1/2. The probability of rolling a 6 and flipping a coin and getting heads is 1/6 × 1/2 = 1/12.
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I need help please ASAP lm really stuck on this question
Answers
Answer:
y=2x+3
Step-by-step explanation:
slope is rise over run. goes up to and over one so your slope is 2/1 or 2.
the c part of the equation of the line is your y-intercept(where the line crosses the y-axis) this is at 3.
so, y=2x+3
A transportation problem with four sources and five destinations will have nine decision variables. True/False
Answers
False. A transportation problem with four sources and five destinations would have 20 decision variables, not nine.
In a transportation problem with four sources and five destinations, the number of decision variables is determined by the number of possible routes from sources to destinations. Each route represents a decision variable, indicating how much flow is sent from a specific source to a specific destination.
For this problem, there would be a maximum of 4 sources and 5 destinations, resulting in a total of (4 * 5) = 20 possible routes. Each route would correspond to a decision variable, indicating the flow from a particular source to a specific destination.
Therefore, a transportation problem with four sources and five destinations would have 20 decision variables, not nine.
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Josie surveyed her classmates. She found 8 students who walk, 30 students who ride the bus to school, 9 who ride bikes, and 3 who ride in cars. What is the experimental probability that the next student Josie surveys will walk to school?
Answers
Answer:
8 out of 50 students will walk.
Step-by-step explanation:
8 + 30 + 9 + 3 = 50
8 students will walk.
8/50
Brainlist pls!
Answer:
About 16%
Step-by-step explanation:
Out of the first 50 students surveyed, 8 of them walk to school. 8/50 is equal to 0.16 or 16% of the students asked. Based off of that sample size, that would be the probability of the next student walking to school.
In a pair of complementary angles, the measure of the larger angle is twice that of the smaller
angle. What is the measure of the larger angle?
Answers
Answer: 60 degrees
Step-by-step explanation:
Pls help me ill give me brainliest
Do all 3 for brainliest pls
Answers
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help meeeeeeeeeeeeee pleaseeeeeeeeeee rn rnnnn!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
help meeeeeeeeeeeeee pleaseeeeeeeeeee rn rnnnn!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Answers
The rocket will take 1.5 seconds to reach its maximum height.
Additionally, the rocket can reach a height of 39 feet.
What is a parabolic function?A function of the form f(x) = ax² + bx + c is called a parabolic function.
The given parabolic function is,
h(t) = -16t²+48t+3 (1),
Initial velocity is 48 feet per second,
And initial height = 3 feet
Differentiate with respect to t,
h'(t)= -32t+48
To find time to substitute this expression to zero.
-32t+48=0
t = 1.5 seconds
For maximum height, substitute this value in equation (1).
h(t) = -16(1.5)²+48(1.5)+3
= -36+72+3
= 39ft
As a result, the rocket achieves its maximum height in 1.5 seconds, reaching a height of 39 feet.
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!!15 POINTS!! What is the range of the function in this table?
Answers
Answer:jmm
Step-by-step explanation:hmmm
Answer:1,3,5
Step-by-step explanation:
is is all your y values
find the equation of the exponential function if the rabbit population has increased to 345 after 1 year.
Answers
The exponential function is P(t)=60(4)^t that can be expressed in terms of feature of the subsequent form: f ( x ) = a x.
Initial population 'a'=60 rabbits
After 1 year, the rabbit population is 240.
An exponential feature is a mathematical feature of the subsequent form: f ( x ) = a x. wherein x is a variable, and a is a regular known as the bottom of the feature.
Equation of the exponential function.
y=ab^x
Put x=0, y=60
60=ab^0
a=60
Put x=1, y=240
240=60(b)^1
b=4
Thus,
y=60(4)^x
P(t)=60(4)^t
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hi i need some help on this, please explain
Answers
Answer:
w = 32
Step-by-step explanation:
Remark
The value of the exterior angle (marked w + 33) = the sum of the two angles not connected to it. w and w + 1
Equation
w + w + 1 = w + 33
Solution
2w + 1 = w + 33 Subtract w from both sides.
2w - w + 1 = 33 Combine
w + 1 = 33 Subtract 1 from both sides.
w = 33 - 1
w= 32
which statistic accurately reflects the vulnerability of prenatal development?
Answers
The statistic that accurately reflects the vulnerability of prenatal development is the incidence of birth defects or congenital anomalies.
Birth defects are structural or functional abnormalities present at birth that can affect various organs or body systems. They can occur during prenatal development due to genetic factors, environmental exposures, or a combination of both. The incidence of birth defects provides an indication of the vulnerability of prenatal development to external influences.
Monitoring and tracking the occurrence of birth defects helps identify potential risk factors, evaluate the impact of interventions or preventive measures, and guide public health efforts. Epidemiological studies and surveillance systems are in place to collect data on birth defects, allowing researchers and healthcare professionals to better understand the causes, patterns, and trends of these conditions.
By examining the prevalence or frequency of birth defects within a population, scientists and healthcare providers can gain insights into the vulnerability of prenatal development and identify areas for targeted interventions, education, and support to minimize the risk and improve the outcomes for prenatal health.
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I give brainliest if it is correct
Answers
Answer:
172 yd²
Step-by-step explanation:
Diante can make 5 neckties in 4 hours.
Select the two true statements about the ratio between the hours he works and the neckties he produces.
A. It takes Diante 1. 25 hours to make one necktie.
B. Diante makes 45 of a necktie per hour.
C. It takes Diante 45 hour to make a necktie.
D. Diante makes 1. 25 neckties per hour
Answers
Answer:
The two true statements about the ratio between the hours Diante works and the neckties he produces are:
A. It takes Diante 1.25 hours to make one necktie.
D. Diante makes 1.25 neckties per hour.
To see this, we can use the formula:
rate = amount of work / time
where "rate" is the number of neckties per hour, "amount of work" is the number of neckties produced, and "time" is the number of hours worked.
Using the given information, we can set up the following equation:
5 / 4 = rate
Solving for the rate, we get:
rate = 1.25 neckties per hour
This means that Diante can make 1.25 neckties in one hour, or it takes him 1.25 hours to make one necktie. Therefore, statements A and D are true. Statements B and C are not true because they do not match the calculated rate.
what is the answer for(-5)(-3)+(-8)-(-17)
Answers
Answer:
Step-by-step explanation:
24
The image of a composite figure is shown.
A four-sided shape with the top side labeled as 9.2 cm. The height is labeled 5 cm. A portion of the base from the perpendicular to a vertex is labeled 3 cm. The portion of the base from the perpendicular to the right vertex is 6.2 cm.
What is the area of the figure?
46 cm2
64.4 cm2
32.2 cm2
39 cm2
Answers
The answer is (D) 39 cm²
What is the formula of a trapezoid?
The formula Area of trapezoid = 1/2 (a + b) h is used to determine the area of a trapezoid, where a and b are the bases (parallel sides) and h is the perpendicular height.
According to figure,
The portion of the base from the perpendicular to the right vertex is 6.2 cm, so we can label the remaining side as 3 + 6.2 = 9.2 cm.
shape is actually a trapezoid, with bases of 9.2 cm and 6.2 cm, and a height of 5 cm. The area of a trapezoid is given by the formula:
Area = (base1 + base2) * height / 2
Substituting the given values, we get:
Area = (9.2 + 6.2) * 5 / 2
Area = 15.4 * 5 / 2
Area = 38.5 cm²
Rounding to the nearest whole number, we get an area of 39 cm². Therefore, the answer is (D) 39 cm².
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find slope (2,2) (3,0)
Answers
Answer:
-2
Step-by-step explanation:
=(0-2)/(3-2)
=-2/1
=-2
PLS PLS PLS THIS IS VOCAB FOR MATH ONLY HAVE UNTILL 7:30 TO DO SEVERAL ASSIGHNMENTS1. a number belonging to the set made up of the counting numbers: 1, 2, 3, and so on
2. a number that is greater than zero
3. a line that graphically represents all numbers
4. a pair of numbers whose sum is zero
5. any number belonging to the set of counting number, or zero
6. a number that is less than zero
7. a number belonging to the set made up of the whole numbers and their opposites
8. two numbers that are the same distance from zero on the number line but in opposite directions
Answers
Answer:
1. natural numbers
2. positive
3. number line
4. zero pair
5. whole number
6. negative
7. integer
8. opposite
Step-by-step explanation:
This is my guess. Take it with a grain of salt. Have a great day!
Answer:
2. positive integers
3. number line
4. I don't know
5. whole number
6. negative integer
7. integers
8. absolute value???
Correct answers only please!
Quincy has $6,000 in a savings account that earns 4% interest, compounded annually.
To the nearest cent, how much interest will he earn in 5 years?
Use the formula B = p(1 + r)t, where B is the balance (final amount), p is the principal (starting amount), r is the interest rate expressed as a decimal, and t is the time in years.
Answers
Answer:
A = $ 7,299.92
A = P + I where
P (principal) = $ 6,000.00
I (interest) = $ 1,299.92
Step-by-step explanation:
A = P(1 + r/n)^nt
Where:
A = Accrued Amount (principal + interest)
P = Principal Amount
I = Interest Amount
R = Annual Nominal Interest Rate in percent
r = Annual Nominal Interest Rate as a decimal
r = R/100
t = Time Involved in years, 0.5 years is calculated as 6 months, etc.
n = number of compounding periods per unit t; at the END of each period
What is the value of 4^6/4^8
A. 1/1024
B. 1/256
C. 1/64
D. 1/16
Answers
Answer:
1/16
Step-by-step explanation:
We know that a^b / a^c = a^(b-c)
4^6/4^8
4^(6-8)
4^-2
We know that a^-b = 1/a^b
1/4^2
1/16
Answer:
D
Step-by-step explanation:
Using the rule of exponents
\(\frac{a^{m} }{a^{n} }\) with n > m
= \(\frac{1}{a^{(n-m)} }\) , then
\(\frac{4^{6} }{4^{8} }\)
= \(\frac{1}{4^{(8-6)} }\)
= \(\frac{1}{4^{2} }\)
= \(\frac{1}{16}\) → D
Suppose f(x)=6x-2 and g(x)=2x+4. Find each of the following functions.
a. (f+g)(x)
b. (f-g)(x)
Answers
Answer:
a) (f + g)(x) = 8x + 2
b) (f - g)(x) = 4x - 6
Step-by-step explanation:
Given:
f(x) = 6x - 2g(x) = 2x + 4a) Find (f + g)(x):
a. (f + g)(x) = f(x) + g(x)
⇒ (f + g)(x) = (6x - 2) + (2x + 4)
⇒ (f + g)(x) = 6x + 2x - 2 + 4
⇒ (f + g)(x) = 8x + 2
b) Find (f - g)(x):
b. (f - g)(x) = f(x) - g(x)
⇒ (f - g)(x) = (6x - 2) - (2x + 4)
⇒ (f - g)(x) = 6x - 2x - 2 - 4
⇒ (f - g)(x) = 4x - 6
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which of the following expressions is equivalent to -10?
a.-7 3
b.-3 - 7
c.3 - 7
d.7 - 3
Answers
The expression which is equivalent to -10 is the option b, -3 - 7.
Explanation:
We can use subtraction and addition of integers to get the value of the given expression. We can write the given expression as;
-3 - 7 = -10 (-3 - 7)
The addition of two negative integers will always give a negative integer. When we subtract a larger negative integer from a smaller negative integer, we will get a negative integer.
If we add -3 and -7 we will get -10. This makes the option b the correct answer.
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4. Write three ratios equivalent to
2:5
Answers
Answer:
4:10, 8:20, 16:40
Step-by-step explanation:
Times both numbers by 2 each time to get a new equivalent ratio.