Evaluate [infinity]∑n=1 1/n(n+1)(n+2). hint: find constants a, b and c such that 1/n(n+1)(n+2) = a/n + b/n+1 + c/n+2.
Answers
the value of the given infinite series is -ln(2) + ∑(n=3 to ∞) 2/n.
What is value?
In mathematics, a value refers to a numerical quantity that represents a specific quantity or measurement.
To evaluate the infinite series ∑(n=1 to ∞) 1/n(n+1)(n+2), we can use the partial fraction decomposition method. As the hint suggests, we want to find constants a, b, and c such that:
1/n(n+1)(n+2) = a/n + b/(n+1) + c/(n+2)
To determine the values of a, b, and c, we can multiply both sides of the equation by n(n+1)(n+2) and simplify the resulting expression:
1 = a(n+1)(n+2) + b(n)(n+2) + c(n)(n+1)
Expanding the right side and collecting like terms:
1 = (a + b + c)\(n^2\) + (3a + 2b + c)n + 2a
Now, we can compare the coefficients of the corresponding powers of n on both sides of the equation:
Coefficients of \(n^2\): 1 = a + b + c
Coefficients of n: 0 = 3a + 2b + c
Coefficients of the constant term: 0 = 2a
From the last equation, we find that a = 0.
Substituting a = 0 into the first two equations, we have:
1 = b + c
0 = 2b + c
From the second equation, we find that c = -2b.
Substituting c = -2b into the first equation, we have:
1 = b - 2b
1 = -b
b = -1
Therefore, b = -1 and c = 2.
Now, we have the decomposition:
1/n(n+1)(n+2) = 0/n - 1/(n+1) + 2/(n+2)
Now we can rewrite the series using the decomposition:
∑(n=1 to ∞) 1/n(n+1)(n+2) = ∑(n=1 to ∞) (0/n - 1/(n+1) + 2/(n+2))
The series can be split into three separate series:
= ∑(n=1 to ∞) 0/n - ∑(n=1 to ∞) 1/(n+1) + ∑(n=1 to ∞) 2/(n+2)
The first series ∑(n=1 to ∞) 0/n is 0 because each term is 0.
The second series ∑(n=1 to ∞) 1/(n+1) is a well-known series called the harmonic series and it converges to ln(2).
The third series ∑(n=1 to ∞) 2/(n+2) can be simplified by shifting the index:
= ∑(n=3 to ∞) 2/n
Now, we have:
∑(n=1 to ∞) 1/n(n+1)(n+2) = 0 - ln(2) + ∑(n=3 to ∞) 2/n
Therefore, the value of the given infinite series is -ln(2) + ∑(n=3 to ∞) 2/n.
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Related Questions
Mr. Nguyen bout a package of 3 chicken legs and a package of 7 chicken wings. Ms. Dawes brought a package of 6 chicken wings. Mr. Nguyen bought 45 ounces of chicken. How much did each chicken leg and each chicken wing weigh?
Answers
Complete Question
Mr. Nguyen bought a package of 3 chicken legs and a package of 7 chicken wings. Ms. Dawes bought a package of 3 chicken legs and a package of 6 chicken wings. Mr. Nguyen bought 45 ounces of chicken. Ms. Dawes bought 42 ounces of chicken. How much did each chicken leg and each chicken wing weigh?
Answer:
Weight of Chicken leg = x = 8 ounces
Weight of Chicken wing = y = 3 ounces
Step-by-step explanation:
Let us represent
Weight of Chicken leg = x
Weight of Chicken wing = y
Mr. Nguyen bought a package of 3 chicken legs and a package of 7 chicken wings.Mr. Nguyen bought 45 ounces of chicken.
3x + 7y = 45 ounces.... Equation 1
Ms. Dawes bought a package of 3 chicken legs and a package of 6 chicken wings. Ms. Dawes bought 42 ounces of chicken.
3x + 6y = 42 ounces... Equation 2
Combining both Equations together we solve by Elimination method
3x + 7y = 45.... Equation 1
3x + 6y = 42... Equation 2
We eliminate x by subtracting Equation 2 from Equation 1
Hence:
y = 3 ounces
Solving for x
3x + 6y = 42 ounces... Equation 2
3x +6(3) = 42 ounces
3x + 18 = 42 ounces
3x =( 42 - 18) ounces
3x = 24 ounces
x = 24 ounces/3
x = 8 ounces
Hence:
Weight of Chicken leg = x = 8 ounces
Weight of Chicken wing = y = 3 ounces
Answer:
chicken leg- 8 oz
chicken wing- 3 oz
Step-by-step explanation:
Please check if these are right if not can someone tell me the correct answer
Answers
Answer: 4 IMAGES
7) slope = 0, Y-intercept = (0, -4)
8) Slope = -1, Y-intercept = (0, -1)
9) Slope= -2/5, Y-intercept= (0, -1)
10) Slope= -2, Y-intercept= (0, 4)
Step-by-step explanation:
I hope it's correct
Which of the following statements is not true about chi-square distributions? The mean decreases as the degrees of freedom increase. OPG? < 0) = 0 O PU2 > 3) is larger for a chi-square distribution with df = 10 than for df = 1 There are an infinite number of chi-square distributions, depending on degrees of freedom. They are always skewed to the right Previous Only saved at 4:44pm
Answers
The statement "The mean decreases as the degrees of freedom increase" is not true about chi-square distributions.
Is it true that the mean of a chi-square distribution decreases as the degrees of freedom increase?In fact, the mean of a chi-square distribution is equal to its degrees of freedom. It does not decrease as the degrees of freedom increase.
The mean remains constant regardless of the degrees of freedom. This is an important characteristic of chi-square distributions.
Regarding the other statements:
The statement "OPG? < 0) = 0" is true. The probability of a chi-square random variable being less than zero is always zero, as chi-square values are non-negative.The statement "OPU2 > 3) is larger for a chi-square distribution with df = 10 than for df = 1" is true. As the degrees of freedom increase, the right-tail probability of a chi-square distribution also increases.The statement "There are an infinite number of chi-square distributions, depending on degrees of freedom" is true. The number of chi-square distributions is infinite because the degrees of freedom can take any positive integer value.The statement "They are always skewed to the right" is generally true. Chi-square distributions tend to be skewed to the right, especially when the degrees of freedom are small.In summary, the statement that is not true about chi-square distributions is that the mean decreases as the degrees of freedom increase.
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the diagram below shows a square-based pyramid
Answers
The solution is, 52 ft is the perimeter of the base of the pyramid.
Here, we have,
Given that:
We have an pyramid with square base.
Area of base of the square pyramid = 169
To find:
Perimeter of the base of pyramid = ?
Solution:
First of all, let us have a look at the formula of area of a square shape.
Area = side * side
Let the side be equal to ft.
Putting the given values in the formula:
169 = a^2
so, a = 13 ft
Now, let us have a look at the formula for perimeter of square.
Perimeter of a square shape = 4 Side
Perimeter = 4 * 13 = 52ft
The solution is, 52 ft is the perimeter of the base of the pyramid.
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complete question:
A pyramid has a square base with an area of 169 ft2. What is the perimeter of the base of the pyramid? A pyramid has a square base with an area of 169 ft2. What is the perimeter of the base of the pyramid?
The quadratic parent function f was transformed to create g(x)=4f(x)which description best represents g?
Answers
Answer: the second one because its a virdacal reflection
Step-by-step explanation: it’s sideways all you have to do is find the equation
what two terms are used to refer to a search in which each value in a sequence is examined until a target value is found or the end of the sequence is reached?
Answers
This problem can be solved using divide and conquer techniques. An method known as "divide and conquer" works by breaking down a large task into smaller ones.
An algorithmic pattern is known as Divide and Conquer. In algorithmic approaches, the idea is to take a dispute involving a large amount of input, divide the input into smaller pieces, solve the problem on each of the smaller pieces, and then combine the piecewise solutions into a comprehensive solution. The Divide and Conquer is the algorithmic name of the approach that was used to solve the issue.
The three steps below are used in a dispute utilizing the Divide and Conquer method.
Create a group of subproblems from the main issue.
Conquer: Recursively and individually solve each subproblem.
Combine: Combine the solutions to the individual subproblems to arrive at the overall solution.
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I WILL GIVE BRAINLYEST
Answers
I used my Graphing calculator and even had my algebra teacher go over it and she said it was correct!
Answer:
0.93
Step-by-step explanation:
Given the function defined in the table below, find the average rate of change, in
simplest form, of the function over the interval 5 ≤ x ≤9.
7
8
9
6
x
4
5
f(x)
4
8
16
32
64
medically explained PAINL X
128
Deltal/ath
Answers
The average rate of change, in simplest form, is -2/5.
What is average rate ?
Divide the change in y-values by the change in x-values to determine the average rate of change. Identifying changes in quantifiable parameters like average speed or average velocity calls for the knowledge of the average rate of change.
The rate of change of the function is its gradient or slope.
The formula for calculating the gradient of a function is expressed as:
\(m=\frac{d y}{d x}=\frac{y_2-y_1}{x_2-x_1}$$\)
Using the coordinate points from the table (0,41) and (15,35)
Substitute the coordinate into the expression:
\($$\begin{aligned}& m=\frac{35-41}{15-0} \\& m=\frac{-6}{15} \\& m=\frac{-2}{5}\end{aligned}$$\)
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(7^5)^7 HURYYYYY................
Answers
Answer:
Exact Form:
\(7^{35}\)
Decimal Form:
3.78818692 x \(10^{29}\)
I hope this helped! :) If this is correct please give me brainliest!
What is the value of ry' when x = 5 and y = 10?
Answers
Answer:
135
Step-by-step explanation:
Answer:
1250
Step-by-step explanation:
X^3 • Y^1
Since X=5 & Y=10,
Substitute
5^3 •10^1
simplify
5*5*5*10=1250
In ΔQRS, s = 860 inches, q = 240 inches and ∠R=144°. Find the length of r, to the nearest inch.
Answers
Answer:
1064
Step-by-step explanation:
Answer: 1064
Step-by-step explanation:
help i dont feel like doing the math
Answers
Answer:
A
Step-by-step explanation:
First, we should put this equation in slope-intercept form by isolating y.
Thus, we have:
\(2x-\frac{4}{3}y=4\\ -\frac{4}{3}y=4-2x\\ y=\frac{3}{2}x-3\)
The slope-intercept equation shows that the y-intercept is -3 and the slope is 3/2.
Only answer A meets these requirements.
you gave your friend a short term 2 year loan of 43.000 at 3% compounded annually. what will be your total return?
Answers
Answer:
45618.7
Step-by-step explanation:
\(f = p \times (1 + r \n) ^{nt} = f \: 43000\)
Prove \(\:\frac{cos\theta }{sec\theta }+\frac{sin\theta }{csc\theta }=1\)
Answers
Answer:
see explanation
Step-by-step explanation:
Using the identities
sec x = \(\frac{1}{cosx} \) , csc x = \(\frac{1}{sinx} \)
sin²x + cos²x = 1
Consider the left side
\(\frac{cos0}{sec0} \) + \(\frac{sin0}{csc0} \)
= \(\frac{cos0}{\frac{1}{cos0} } \) + \(\frac{sin0}{\frac{1}{sin0} } \)
= cos²Θ + sin²Θ
= 1 = right side , thus proven
what is the surface area of a conical grain storage tank that has a height of 42 meters and a diameter of 24 meters
Answers
The surface area of a conical grain storage tank that has a height of 42 meters and a diameter of 24 meters is 1,670.5 square meters.
The surface area of a cone is calculated using the following formula:
Surface area = πr² + πrl, where π is approximately equal to 3.14, r is the radius of the base of the cone, and l is the slant height of the cone.
In this problem, we are given that the height of the cone is 42 meters and the diameter of the base is 24 meters. The radius of the base is half of the diameter, so the radius is 12 meters. The slant height of the cone can be calculated using the Pythagorean theorem.
l² = 12² + 42²
l² = 1764
l = 42.01 meters
The surface area of the cone is then calculated as follows:
Surface area = πr² + πrl
Surface area = 3.14 * 12² + 3.14 * 12 * 42.01
Surface area = 1,670.5 square meters
Here are some additional explanations:
The radius of a circle is the distance from the center of the circle to any point on the edge of the circle.The slant height of a cone is the distance from the vertex of the cone to any point on the edge of the base of the cone.The surface area of a cone is calculated by adding the area of the base of the cone and the area of the lateral surface of the cone.The area of the base of a cone is calculated using the formula πr².The area of the lateral surface of a cone is calculated using the formula πrl.To know more about area click here
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can someone help plzzz asappp :///
Answers
Answer:
I believe answer choice A
Step-by-step explanation:
Answer:
I thin that the answer is A
Step-by-step explanation:
The twelve-inch square tiles are shipped in boxes of 20 pieces per box. Each of the boxes weighs 36 pounds. Approximately how many ounces does each tile weigh?
Answers
Each twelve-inch square tile weighs approximately 27 ounces.
To calculate the weight of each tile in ounces, we need to convert the weight of the box from pounds to ounces and divide it by the number of tiles in the box. Since there are 16 ounces in a pound, the weight of each box is 36 pounds * 16 ounces/pound = 576 ounces.
If there are 20 tiles in each box, we divide the weight of the box (576 ounces) by the number of tiles (20) to get the weight of each tile: 576 ounces / 20 tiles = 28.8 ounces. Rounding to the nearest ounce, each twelve-inch square tile weighs approximately 27 ounces.
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4x – 3 = 2x +9
solve for x
Answers
Answer:
6
Step-by-step explanation:
Add three to both sides
Subtract 2x from both sides
2x=12
Divide by 2
X=6
Answer:
x = 6
Step-by-step explanation:
4x - 3 = 2x + 9
2x = 12
x = 6
I need help fast plss
Answers
The measures of the angles are ∠1 = 127°, ∠2 = 53°, ∠3 = 127°, ∠4 = 37°, ∠5 = 53°, ∠6 = 90°, ∠7 = 37°, ∠8 = 143°, ∠9 = 37° and ∠10 = 143°
Finding the measures of the anglesFrom the question, we have the following parameters that can be used in our computation:
The transversal lines and the other lines
So, we have
∠1 = 180 - 53
Evaluate
∠1 = 127°
Also, we have
∠5 = 53°
By vertical angles, we have
∠2 = 53°
∠3 = 127°
Next, we have
∠4 = 127 - 90°
∠4 = 37°
Solving further, we have
∠6 = 90°
By corresponding angles, we have
∠7 = 37°
∠9 = 180 - 90 - 53°
∠9 = 37°
∠10 = 90 + 53°
∠10 = 143°
∠8 = 143°
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simplify the ratio.
36:81:27
Answers
Answer:
4:9:3
Step-by-step explanation:
divide by nine
Hope this helps
Answer:
4:9:3
Step-by-step explanation:
36 / 9 = 4
81 / 9 = 9
27 / 9 = 3
Which polynomial is prime? x2 x 1 x2 – x – 2 x2 – 12x 11 x2 2x – 8.
Answers
Answer: x^2 + x + 1
.....
if the first table in a cartesian join has five rows and the second table has three rows, the results will consist of ____________________ rows.
Answers
In a Cartesian join, also known as a cross join, each row from the first table is combined with each row from the second table.
If the first table has 5 rows and the second table has 3 rows, the result of the Cartesian join will consist of 5 * 3 = 15 rows.
Frozen yogurt costs $0.52 per ounce. What is the cost of 8 ounces of frozen yogurt?
$
Answers
Answer:
$4.16
Step-by-step explanation:
To find out how much 8 ounces of frozen yogurt costs if each ounce costs $0.52, we can use an equation that looks like this:
0.52 × 8 = 4.16Therefore, for 8 ounces of yogurt is will cost $4.16.
T/F
Approximately 99.7 percent of sample means will fall within plus or minus two standard deviations of the process mean if the process is under control.
Answers
This statement is True. The statement is known as the 68-95-99.7 rule, which applies to a normal distribution. According to this rule, approximately 99.7 percent of the sample means will fall within plus or minus two standard deviations of the process mean if the process is under control. This rule is derived from the properties of a normal distribution and provides a guideline for assessing the variability of a process.
In statistical terms, if a process is under control and follows a normal distribution, about 68 percent of the sample means will fall within one standard deviation of the process mean, around 95 percent will fall within two standard deviations, and roughly 99.7 percent will fall within three standard deviations. This rule assumes that the process is stable and does not have any significant shifts or outliers.
The 99.7 percent figure is derived from the empirical rule, which is based on the characteristics of a normal distribution. It is widely used in statistical process control and quality management to assess the performance and variability of a process. By monitoring the sample means and their deviation from the process mean, practitioners can identify if the process is in control or if there are any significant variations that need to be addressed.
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What do you think the narrator means at the end of the story when she says that she too has planted marigolds?
Answers
The narrator suggests that she now lives her life by striving to find optimism in even the worst circumstances when she adds, "I too have planted marigolds," at the conclusion of the story.
Because the narrator characterized the marigolds in the story as "Dazzling strips of bright petals bunch together in gigantic mounds warm and passionate in sun golden," the narrator's description of and response to seeing the marigolds is rather favorable.
The ability to comprehend and appreciate others as fully human beings serves as the story's central subject and marks the start of adulthood. Lizabeth felt ashamed when she realized she had wronged a genuine human being instead of a "witch" when she vented her anger at the marigolds and turned to look at Miss Lottie.
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helpppppppppp plsssssss
Answers
Answer: I think it might be the one u chose but im not sure but if i got it wrong then im sorry and feel free to correct me
Step-by-step explanation:
Answer:
parallal
neither
perpendicular
Step-by-step explanation:
Hopes this helps! : D
And I am so sorry if I get you the wrong answers.
What should the bottom right stamp be numbered with?
Answers
Observe each pairs
#1
(82,4)
4×2=8#2
(93,3)
3×3=9#3
(61,6)
6×1=6#4.
(22,?)
1×2=2Answer is 1Answer:
• let the upper row be y and lower row be x
• Considering (4, 82):
\({ \rm{x :y = 4 :82 }} \\ \\ { \rm{ \frac{x}{y} = \frac{4}{82} }} \\ \\{ \boxed { \rm{ \: x = \frac{4y}{82} \: }}}\)
• when y is 22:
\({ \rm{x = \frac{(4 \times 22)}{82} }} \\ \\ { \rm{x \approx1}}\)
Answer: It'd be 1
Find the length of side BC.
Give your answer to 3 significant figures.
C
A
71°
6.3 cm
B
Answers
From the given diagram, the size of side BC is 19.4 rounded to 3 significant figures
What is trigonometric function?The trigonometric functions are real functions that relate the angle of a right-angled triangle to side length ratios. They are widely used in all geosciences, including navigation, solid mechanics, celestial mechanics, geodesy, and many others.The values of all trigonometric functions based on the ratio of sides in a right-angled triangle are defined as trigonometric ratios. The trigonometric ratios of any acute angle are the ratios of the sides of a right-angled triangle with respect to that acute angle.From the given diagram, the size of side BC is 19.4 rounded to 3 significant figures
Given
The right angle triangle with a side and an angle
We must find out the hypotenuse BC
We are given with adjacent side AB
Let's use trigonometric ratio
\($&\cos (B)=\frac{\text { adjacent }}{\text { hypotenuse }} \\\)
\($&\cos (71)=\frac{A B}{B C} \\\)
\($&\cos (71)=\frac{6.3}{B C} \\\)
\($&B C \cos (71)=6.3 \\\)
\($&B C=\frac{6.3}{\cos (71)}\)
Use the calculator to find out BC
BC=19.3508
Round the answer to 3 significant figures
So BC= 19.4
Length of side BC = 19.4
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o is the midpoint of LN
definition of
Answers
Answer:
Difficult to infer an answer with such little information but I would assume it's "Definition of Midpoint" if you're writing a proof.
A television station shows 3 commercials every 12 minutes. At that rate, how many commercials will the station show in 60 minutes?
Answers
Answer:
zqxwsecdvrfr
Step-by-step explanation: