Answers
By the ratio test, the series converges for
\(\displaystyle \lim_{k\to\infty} \left|\frac{(x+2)^{k+1}}{\sqrt{k+1}} \cdots \frac{\sqrt k}{(x+2)^k}\right| = |x+2| \lim_{k\to\infty} \sqrt{\frac k{k+1}} = |x+2| < 1\)
so that the radius of convergence is 1, and the interval of convergence is
|x + 2| < 1 ⇒ -1 < x + 2 < 1 ⇒ -3 < x < -1
The radius of convergence R is 1 and the interval of convergence is (-3, -1) for the given power series. This can be obtained by using ratio test.
Find the radius of convergence R and the interval of convergence:Ratio test is the test that is used to find the convergence of the given power series.
First aₙ is noted and then aₙ₊₁ is noted.
For ∑ aₙ, aₙ and aₙ₊₁ is noted.
\(\lim_{n \to \infty} |\frac{a_{n+1}}{a_{n} }|\) = β
If β < 1, then the series convergesIf β > 1, then the series divergesIf β = 1, then the series inconclusiveHere \(a_{k}\) = \(\frac{(x+2)^{k}}{\sqrt{k} }\) and \(a_{k+1}\) = \(\frac{(x+2)^{k+1}}{\sqrt{k+1} }\)
Now limit is taken,
\(\lim_{n \to \infty} |\frac{a_{n+1}}{a_{n} }|\) = \(\lim_{n \to \infty} |\frac{(x+2)^{k+1} }{\sqrt{k+1} }/\frac{(x+2)^{k} }{\sqrt{k} }|\)
= \(\lim_{n \to \infty} |\frac{(x+2)^{k+1} }{\sqrt{k+1} }\frac{\sqrt{k} }{(x+2)^{k}}|\)
= \(\lim_{n \to \infty} |{(x+2) } }{\sqrt{\frac{k}{k+1} } }}|\)
= \(|{x+2 }|\lim_{n \to \infty}}{\sqrt{\frac{k}{k+1} } }}\)
= \(|{x+2 }|\) < 1
- 1 < \({x+2 }\) < 1
- 1 - 2 < x < 1 - 2
- 3 < x < - 1
We get that,
interval of convergence = (-3, -1)
radius of convergence R = 1
Hence the radius of convergence R is 1 and the interval of convergence is (-3, -1) for the given power series.
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Related Questions
help me please , for 20 points
Answers
Answer:
A
Step-by-step explanation:
First off we can see that there are 46 pencils - and 7 students - so we can divide the two
46/7 is not a whole number, so we need to find a number that can be multiplied by 7, and is right below 46.
If you multiply 7 by 6, you get 42, which is the highest you can get with multiply by 7, which is also below 46.
Now that we know that each student gets 6 pencils, we need to find out how many are left.
We can do 46 (total pencils) minus 42 (pencils given) to get a leftover of 4 pencils, so the answer is A.
Determina el valor de un número negativo tal que la suma de
ese número y su cuadrado es 42.
Answers
The value of a negative number such that the sum of that number and its square is 42 is -7.
What are negative numbers?A negative number is one that always has a value lower than zero and is denoted by the minus (-) symbol. On a number line, negative numbers are displayed to the left of zero. Negative numbers include -6 and -15 as examples.Let the negative number be -x.
According to the question, the sum of the negative number and its square is 42.
So, we form an equation
- x + (-x)² = 42
x² - x - 42 = 0
Solving the quadratic equation,
x² - 7x + 6x - 42 = 0
x(x - 7) + 6(x - 7) = 0
(x - 7)(x + 6) = 0
x = -6 and x = 7
We have taken the number -x to be negative.
So, -x = -6 and -x = 7
x = 6 and x = -7
Hence, the value of the negative number is -7.
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The correct question is: "Determine the value of a negative number such that the sum of that number and its square is 42."
The large rectangle below represents one whole.
What percent is represented by the shaded area?
Answers
Answer:
40%
Step-by-step explanation:
so have 5 rectangles in a whole.we also have 2 shaded potions in that whole which represents 2/5 shaded potions.2/5 in fraction =0.4Now Change 0.4 to percentage by multiplying 0.4 by 1000.4 x 100 = 40Therefore the percentage represented by the shaded area is 40% Hope this helps.Good luck ✅.Milo can run 12 miles in 60 minutes. He needs to reduce his time by 19%. Approximately how many minutes does he have to take off his time? Round to the nearest whole number.
Answers
Milo needs to cover 12 miles in 48.6 minutes, as the percentage.
What is percentage?
A percentage that represents a tenth of a quantity. One percent, denoted by the symbol 1%, is equal to one-hundredth of something; hence, 100 percent denotes the full thing, and 200 percent designates twice the amount specified. A portion per hundred is what the percentage denotes. The percentage refers to one in a hundred. The % sign is used to denote it.
Milo can run 12 miles in 60 minutes.
She needs to reduce the time by 19%
So we have to calculate the 19%of 60
= 19/100* 60
=57/5 minutes
= 11.4
So the time he needs to achieve will be 60-11.4 = 48.6 minutes.
Hence, he needs to cover 12 miles in 48.6 minutes.
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n a bolt factory, machines a, b, and c manufacture 25%, 35%, and 40% of the total of their output, respectively. out of them, 5%, 4%, and 2% are defective bolts. a bolt is drawn at random from the product and is found to be defective. what are the probabilities that it was manufactured by machines a, b, and c?
Answers
The probabilities that the defective bolt was manufactured by machines A, B, and C are approximately 0.3623, 0.4058, and 0.2319, respectively.
To solve this problem, we can use Bayes' theorem.
Let's denote the events as follows:
A: Bolt is manufactured by machine A
B: Bolt is manufactured by machine B
C: Bolt is manufactured by machine C
D: Bolt is defective
We need to find the conditional probabilities P(A|D), P(B|D), and P(C|D). According to Bayes' theorem:
P(A|D) = (P(D|A) x P(A)) / P(D)
P(B|D) = (P(D|B) x P(B)) / P(D)
P(C|D) = (P(D|C) x P(C)) / P(D)
We are given the following information:
P(A) = 0.25 (machine A manufactures 25% of the total output)
P(B) = 0.35 (machine B manufactures 35% of the total output)
P(C) = 0.40 (machine C manufactures 40% of the total output)
P(D|A) = 0.05 (5% of machine A's output is defective)
P(D|B) = 0.04 (4% of machine B's output is defective)
P(D|C) = 0.02 (2% of machine C's output is defective)
To calculate P(D), we can use the law of total probability:
P(D) = P(D|A) x P(A) + P(D|B) x P(B) + P(D|C) x P(C)
Let's substitute the given values into the equations:
P(D) = (0.05 x 0.25) + (0.04 x 0.35) + (0.02 x 0.40)
= 0.0125 + 0.014 + 0.008
= 0.0345
Now, we can calculate the conditional probabilities:
P(A|D) = (0.05 x 0.25) / 0.0345
= 0.0125 / 0.0345
≈ 0.3623
P(B|D) = (0.04 x 0.35) / 0.0345
= 0.014 / 0.0345
≈ 0.4058
P(C|D) = (0.02 x 0.40) / 0.0345
= 0.008 / 0.0345
≈ 0.2319
Therefore, the probabilities that the defective bolt was manufactured by machines A, B, and C are approximately 0.3623, 0.4058, and 0.2319, respectively.
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HELPPP MEEE PLEASE BE QUICK AS SOON AS POSSIBLE
Answers
Add the frequency’s together:
84 + 36 + 72 + 84 + 204 = 480
Divide each frequency by the total then multiply by 360 degrees:
A = 84/480 x 360 = 63 degrees
B = 36/480 x 360 = 27 degrees
C = 72/480 x 360 = 54 degrees
D = 84/480 x 360 = 63 degrees
E = 204/480 x 360 = 153 degrees
Consider equation (1) again, ln (wage) = β0 + β1 educ + β2 exper + β3 married + β4 black + β5 south + β6 urban +u
(a) Explain why the variable educ might be endogenous. How does this affect the estimated coefficients? Does the endogeneity of educ only affect the estimate of β2 or does it affect the coefficients associated with other variables?
(b) The variable brthord is birth order (one for the first-born child, two for a second-born child and so on). Explain why brthord could be used as an instrument for educ in equation (1). That is, does this variable satisfy the relevance and exogeneity conditions for it to be an appropriate instrument?
Answers
(a) The variable educ might be endogenous
(b) The variable brthord is birth order (one for the first-born child, two for a second-born child and so on) could be used as an instrument for educ in equation
a) The variable instruction might be endogenous because as compensation increases the income expansions which additionally make able to an individual more educating himself. So there is an opportunity for the instruction might be an endogenous variable.
The indigeneity may involve the 32 the coefficient of knowledge as well different variables like married, black, south, urban, etc.
b) There is a substantial high relationship exists between birth order and the status of teaching. it is more possible to have higher schooling with less the order of child-born and the birth order is autonomous of the error term as well with wage. So the variable "birth order" is a good variable to use as an agency for the endogenous variable instruction.
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very fast
Show, by induction, that \( T(n)=10 n^{2}-3 n \quad \) if \( n=1 \)
Answers
Given that \(\(T(n)\) = \(10n^2-3n\)\) if (\(\(n=1\)\)), you have to prove it by induction. So, we have proved it by induction that \($$\(T(n)=10n^2-3n\)$$\) if ( n= 1). The given statement is true for all positive integers n
Let's do it below: The base case (n=1) is given as follows: \(T(1)\) =\(10\cdot 1^2-3\cdot 1\\&\)=\(7\end{aligned}$$\). This implies that \(\(T(1)\)\) holds true for the base case.
Now, let's assume that \(\(T(k)=10k^2-3k\)\) holds true for some arbitrary \(\(k\geq 1\).\)
Thus, for n=k+1, T(k+1) = \(10(k+1)^2-3(k+1)\\&\) = \(10(k^2+2k+1)-3k-3\\&\)=\(10k^2+20k+7k+7\\&\) = \(10k^2-3k+20k+7k+7\\&\) = \(T(k)+23k+7\\&\) = \((10k^2-3k)+23k+7\\&\) = \(10(k+1)^2-3(k+1)\).
Therefore, we have proved that the statement holds true for n=k+1 as well. Hence, we have proved it by induction that \($$\(T(n)=10n^2-3n\)$$\) if (n=1). Therefore, the given statement is true for all positive integers n.
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Secants ⎯⎯⎯⎯⎯⎯⎯⎯⎯ and ⎯⎯⎯⎯⎯⎯⎯⎯⎯ intersect in the exterior of a circle. ∠ measures 27.42° and the measure of DE is 73.18°. Determine the measure of BC.
Answers
Answer:
BC = 18.34°
Step-by-step explanation:
Recall: Exterior angle formed by 2 secants = ½(the difference of the measures of the intercepts arcs)
Thus:
m<BAC = ½(DE - BC)
27.42° = ½(73.18 - BC) => Substitution
Multiply both sides by 2
2*27.42 = 73.18 - BC
54.84 = 73.18 - BC
BC = 73.18 - 54.84
BC = 18.34°
what is the doamin of the function f(x)=x+1/x²-6x+8
Answers
Answer:
Step-by-step explanation:
(x+1) = domain = all real #s
(x²-6x-8)⇒ (x-4)(x-2)= x≠ 2,4
Because you can not have a zero in the denominator, x≠2,4
so your domain is (-∞,2)∪(2,4)∪(4,∞)
Consider this triangle.
Answers
use sin
sin(50) = x/6
sin(50) x 6 = x
x = ~4.5963
An irrational number between 5/7 and 7/9 is?
Answers
Answer:
Step-by-step explanation:
-----------
5/7 can be written as 0.714285
-----
7/9 can be written as 0.777
∴ An irrational number between 5/7 and 7/9 is 0.72010010001.....
or u ca write other numbers also like
0.73010203.....
0.74090080007.....
20 POINTS! HURRY!!!!Which expressions are equivalent to 5(1/3x+7)-3(1/3x-4)? Select three options.5 1/3x-3 1/2x+35-121/6x+471 2/3x+36-1 1/2x+ 125(1/3x)+(5)(7)-(3)(1/2x)+(3)(4)1 1/3x+35-1 1/2x-12
Answers
The given expression is:
\(undefined\)A significance test about a proportion is conducted using a significance level of 0.05. The sample statistic is 0.12. The p-value is 0.03.
a) If H0 were true, for what probability of a Type I error was the test designed?
b) What conclusion (reject or fail to reject) would you make for this test?
c) If this test resulted in a decision error, what type of error was it?
Answers
With a significance test about a proportion is conducted using a significance level of 0.05, sample statistic is 0.12, p-value is 0.03.
a) If H0 were true, the test was designed for a probability of a Type I error of 0.05.
b) Based on the p-value of 0.03, you would reject the null hypothesis.
c) If this test resulted in a decision error, it would be a Type II error.
a) If H0 were true, the test was designed for a probability of a Type I error of 0.05. When conducting a hypothesis test, we set a significance level, often denoted by alpha (α), to control the probability of making a Type I error.
A common value for alpha is 0.05, which means that we are willing to accept a 5% chance of making a Type I error.
b) Based on the p-value of 0.03, which is less than the significance level of 0.05, you would reject the null hypothesis. This means that the sample statistic provides enough evidence to suggest that the population proportion is different from the hypothesized value.
c) If this test resulted in a decision error, it would be a Type II error. A Type II error occurs when you fail to reject a false null hypothesis. In this case, the p-value of 0.03 suggests that the population proportion is different from the hypothesized value, so if the null hypothesis were actually false and you still failed to reject it, it would be a Type II error.
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Find the GCF of 4 and 20 and please list the factors by the number..
Answers
Answer:
4 20
1-4 1-20
2-2 2-10
4-5
Step-by-step explanation:
Please mark brainliest
The figures are similar. Find the missing length.Round to the nearest thousandth
Answers
Answer:
The missing x = 0.675 m
Step-by-step explanation:
In similar triangles, the corresponding sides are proportion (means their ratios are equal)
In the given figure
∵ The two triangles are similar
∴ Their corresponding sides are proportion
∵ The sides of lengths 1.8 m and x are corresponding sides
∵ The sides of lengths 8.0 m and 3.0 are corresponding sides
∴ \(\frac{1.8}{x}\) = \(\frac{8.0}{3.0}\)
→ By using cross multiplication
∴ x × 8.0 = 1.8 × 3.0
∴ 8x = 5.4
→ Divide both sides by 8
∵ \(\frac{8x}{8}\) = \(\frac{5.4}{8}\)
∴ x = 0.675
→ No rounding because there is no digit after the thousandth digit
∴ x = 0.675 m
a careless university student leaves her iclicker device behind with probability 1/4 each time she attends a class. she sets out with her iclicker device to attend 5 different classes (each class is in a different lecture theatre). part 1) if she arrives home without her iclicker device and she is sure she has the iclicker device after leaving the first class, what is the probability (to 3 significant figures) that she left it in the 5th class? probability
Answers
The probability of leaving it in any given class is 1/4, the probability of leaving it in the 5th class is 1/4, or\(0.250 (25.0%).\)
The probability that the student left her iClicker device in the 5th class is 0.250 (25.0%). This is because the probability of leaving it in any given class is 1/4, and as she attends 5 different classes,
there is 5/4 or 1.25 chances of her leaving it in the 5th class.
To explain further, the probability of an event can be calculated by dividing the number of ways the event can occur by the total number of possible outcomes. In this case, the event is leaving the iClicker device in the 5th class, and the total number of possible outcomes is 5 (for the 5 classes).
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The length of a rectangle is twice its width. If the area of the rectangle is 162in^2, find its perimeter.
Answers
We know that the length of a rectangle is twice its width. Then:
The area of a rectangle is the product of its width and length. Then:
\(A=2x\cdot x=2x^2\)Additionally, we know that the area is 162 in². Using the expression above we can obtain the value of x:
\(\begin{gathered} 162=2x^2 \\ 81=x^2 \\ x=9\text{ in} \end{gathered}\)Finally, the perimeter (2P) is just twice the sum of the width and the length of the rectangle:
\(2P=2\cdot(2x+x)=6x=6\cdot9=54\text{ in}\)
Sergio runs 2 and 1/4 miles in 30 minutes. What is Sergio's speed in miles per hour ? Answer as a decimal.
Answers
Answer:
4.5 miles per hour
Step-by-step explanation:
2 1/4 x 2 = 4.5 (As decimal)
30 mins + 30 mins = 1 hour
1 hour = 4.5 miles per hour
Highschool Multi-Step Equations
Answers
Answer:
y=-3
Step-by-step explanation:
I hope this works
How to tell something is a function
Answers
Answer: Something is a function if the same x value has different y values.
An example of a nonfunction would be a function containing the coordinates (1,5) and (1,6)
Step-by-step explanation:
What does the graph of the parametric equations x(t)=3−t and
y(t)= (t+1)^2 , where t is on the interval [−3,1], look like? Drag
and drop the answers to the boxes to correctly complete the
statemen
The parametric equations graph as a portion of a parabola. The initial point is and the terminal point is The vertex of the parabola is Arrows are drawn along the parabola to indicate motion right to
Answers
The parametric equations graph as a portion of a parabola. The initial point is (3, 4) and the terminal point is (2, 4). The vertex of the parabola is at (2, 4). Arrows are drawn along the parabola to indicate motion from right to left.
The graph of the parametric equations \(x(t) = 3 - t\) and y(t) =\((t + 1)^2\), where t is on the interval [-3, 1], represents a portion of a parabola. The initial point of the graph is \((3, 4)\) when \(t = -3\), and the terminal point is (2, 4) when t = 1. The vertex of the parabola occurs at \((2, 4)\), which is the lowest point on the curve. As t increases from \(-3 \ to \ 1\), the x-coordinate of the points decreases, indicating a right-to-left motion along the parabola. The parabola opens upwards, creating a concave shape. The graph displays the relationship between x and y values as t varies within the given interval.In conclusion, the parametric equations graph as a portion of a parabola. The initial point is (3, 4) and the terminal point is (2, 4). The vertex of the parabola is at (2, 4). Arrows are drawn along the parabola to indicate motion from right to left.
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y = mx + b -2 = 3(4) + b -2 = 12 + b -14 = b; the y-intercept is -14 What is wrong with this answer?
Answers
Answer:
There is nothing wrong with the answer obtained
Step-by-step explanation:
Given the equation :
Y = mx + b
-2 = 3(4) + b
-2 = 12 + b
-14 = b
The correct calculation :
Y = mx + b
-2 = 3(4) + b
-2 = 12 + b
Subtract 12 from both sides
-2 - 12 = 12 + b - 12
-14 = b
Based on the details of the question given, the intercept, b is - 14 ;
Hence, there is nothing wrong with the answer
While shopping kyla found a dress that she would like to purchase but it cost $52.25 mor3 than she has
Answers
Kyla should babysit for 9.5 hours to buy the dress.
Double number line represents the comparison of numbers by representing them in equal ratio. The required double number line for the question is attached as figure.
The row I indicates the amount earned and row II indicates the number of hours Kyla should babysit. As is evident from the double number line, the average of last two amounts is the amount required by Kyla. Calculations are as follows -
Required amount = (49.5 + 55)/2
Required amount = 52.25
Now, the time between last two columns of row II will also be taken as average for calculation. So, time required to earn 52.25 after earning 49.5: (9 + 10)/2
Time = 0.5
Total time required to babysit = 9 + 0.5
Total time = 9.5 hours
Thus, Kyla should babysit for 9.5 hours to buy the dress.
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The complete question is -
While shopping, Kyla found a dress that she would really like, but it cost $52.25 more than she has. Kyla charges $5.50 an hour for babysitting. She wants to figure out how many hours she must babysit $52.25 to buy the dress. Use a double number line to support your answer.
The number of fish in a lake is decreasing by 400 every year, as described by the function below.
f
(
1
)
=
8200
f
(
n
+
1
)
=
f
(
n
)
−
400
What will be the value of
f
(
5
)
?
Answers
Using the recursive function given, it is found that f(5) = 6600.
The function given is:
\(f(n + 1) = f(n) - 400\)
\(f(1) = 8200\)
To find f(5), we keep applying the function until \(n + 1 = 5\), hence:
f(2) is f(1) subtracted by 400
\(f(2) = f(1) - 400 = 8200 - 400 = 7800\)
f(3) is f(2) subtracted by 400
\(f(3) = f(2) - 400 = 7800 - 400 = 7400\)
f(4) is f(3) subtracted by 400
\(f(4) = f(3) - 400 = 7400 - 400 = 7000\)
f(5) is f(4) subtracted by 400
\(f(5) = f(4) - 400 = 7000 - 400 = 6600\)
Hence, the result is f(5) = 6600.
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Three times a number increased by
twice the number is greater than
125. What is the number?
Answers
Answer:
x > 25
Step-by-step explanation:
This question relates to algebra. We can solve this problem by creating a variable for the solution (which we will find). We will link the variable in an equation l with the information given.
So let's translate each piece of information mathematically. First, the question tells us "Three times a number." The first part uses multiplication (times) linked to our number (we will use x for our variable). So we write:
3x
The second part is "increased by twice the number" Increasing is adding (+) so we add two times the number. So:
3x + 2x
The final part says "is greater than 125" so we need to use the > sign
3x + 2x > 125
Now we can solve by collecting like terms and moving them to a different side.
5x > 125
x > 25
PLEASE HELP ASAP, THANK YOU!
Answers
Answer:
Your answer is \(10x + 5\)
Step-by-step explanation:
A coefficient is the number before the variable, in this case, 10.
A constant is the number with no variable attached to it, in this case, 5.
PLEASE HELP!
Which statement is true about days and minutes?
One day is 60 times as long as a minute.
One minute is 60 times as long as a day.
One day is 1,440 times as long as a minute.
One minute is 1,440 times as long as a day.
Answers
Answer:
D
Step-by-step explanation:
Which of the following tables represents a function?
A) x 2 2 3 3
y 5 -5 6 -6
B) x 4 -4 4 -4
y 1 2 3 4
C) x 6 6 7 7
y 2 3 4 5
D) x 2 -2 4 -4
y 8 8 12 12
Answers
Answer:
B
Step-by-step explanation:
Every time the y-value increases by one, the function multiplies the x-value by -1
Answer:
Option D
Step-by-step explanation:
cause in function one single domain can have many range but many range can't have single domain.
of girls who tried out for the lacrosse team at euclid middle school, 12 were selected. of the 40 boys who tried out , 16 were selected are the ratios of the number of students on the team to the number of students trying out the same for both boys and girls how do you know
Answers
Answer: Yes, is the same ratio.
Explanation:
For the girls we have that out of the 30 that tried out for the team, 12 were selected. The ratio for the girls is:
\(\frac{12}{30}=\frac{2}{5}\)(we simplify 12/30 by dividing the numerator and denominator by 6)
For the boys we have that out of the 40 that tried out for the team, 16 were selected. The ratio for the boys is:
\(\frac{16}{40}=\frac{2}{5}\)(we simplify 16/40 by dividing the numerator and denominator by 8)
We can see that the ratio of the number of students on the team to the number of students trying out is the same for girls and boys.