The given series converges by the alternating series test, and the correct answer is A, "This series converges by the Alternating Series Test."
To use the alternating series test, we need to check two conditions:
The sequence [tex](1/n^2)[/tex] is decreasing and approaches zero as n approaches infinity.
The terms of the series alternate in sign and decrease in absolute value.
Let's check the first condition:
lim (n→∞) n/[tex](n^2+2)[/tex] = 0
To see this, note that as n becomes very large, [tex]n^2+2[/tex] grows much faster than n, so [tex]n/(n^2+2)[/tex] approaches zero as n approaches infinity. Therefore, the first condition is satisfied.
Next, let's check the second condition:
0 ≤ 1/(n+2) ≤ [tex]n/(n^2+2)[/tex] for n ≥ 1
To see this, note that for n ≥ 1, we have:
1/(n+2) ≤ [tex]n/(n^2+2)n/(n^2+2)[/tex]
Multiplying both sides by [tex](-1)^{(n-1)[/tex] and summing over all n, we get:
[tex]\sum n=1 \infty^{(n-1)} (1/(n+2)) $\leq$ \sum n=1infinity^{(n-1)}(n/(n^2+2))[/tex]
Since the series on the right-hand side is the given series, and the series on the left-hand side is the alternating harmonic series, which is known to converge, the second condition is also satisfied.
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To determine whether the given series is convergent or divergent, we need to use the alternating series test. For this, we need to show that the terms of the series are decreasing in absolute value and that the limit of the terms as n approaches infinity is zero.
In this case, we need to show that Lim n [infinity] n/(n^2+2) = 0 and that O ≤ 1/(n+2) ≤ n/n²+2 for 1≤n. After verifying these conditions, we can conclude that the given series converges by the Alternating Series Test. Therefore, option A is the correct answer. The divergence test is not applicable here, as the series alternates between positive and negative terms. Thus, option B is incorrect. The convergence test is conclusive in this case, and option C is also incorrect.
We are given the series ∑n=1 to infinity (−1)^(n−1)(n/(n^2+2)). To apply the Alternating Series Test (AST), we need to check two conditions:
1. Lim n→infinity (n/(n^2+2)) = 0
2. The sequence n/(n^2+2) is non-increasing and positive for n≥1
1. To find the limit, divide both numerator and denominator by n^2:
Lim n→infinity (n/(n^2+2)) = Lim n→infinity (1/(1+(2/n^2))) = 1/1 = 0
2. The inequality 0 ≤ 1/(n+2) ≤ n/(n^2+2) can be rewritten as 0 ≤ 1/(n+2) ≤ 1/(1+2/n), which is true for n≥1.
Since both conditions are satisfied, the series converges by the Alternating Series Test (AST). Therefore, the correct answer is A.
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someone please help mee (will give brainliest 20 points!!!)
Step-by-step explanation:
the functional graph is the line.
a functional graph is always showing the direct relationship between a value on one coordinate axis with a value in the other coordinate axis
here it is between the price in dollars (as we can understand it : per unit) and the number of units sold.
clearly, the more units we sell, the cheaper we have to make the price per unit.
the question is how many units can be sold for a given price ($60).
so, I find the given price on the price axis (in this case the vertical axis), go then along that price all the way to the right until I hit the function line, and then go straight down to the unit axis and see what value I am hitting there : 80. which means 80,000 units.
so, I can sell 80,000 units for $60 per unit.
Representing a Graph with a Slope-Intercept Equation
-5-4-3-2
5
4
3
2
+
2345
y
2 3 4
X
5
A function f(x) is graphed.
What is the slope of the function?
m
What is the y-intercept of the function?
b = v
Which equation represents the graph of the function?
Answer:
1.) The slope is 2
2.) The y-intercept is (0,1)
3.) y = 2x + 1
Step-by-step explanation:
1.) you can either use rise over run or the slope formula. rise over run is quicker, but here's the formula:
you can use any two points on the line, i chose (2, 5) and (1, 3)
[tex]\frac{(y_{2}-y_{1})}{(x_{2}-x_{1})} \\[/tex]
[tex]\frac{3-5}{1-2} \\\\\frac{-2}{-1} \\\\\frac{2}{1} \\\\2[/tex]
2.) for the y-intercept, you just look for the point where the graph crosses the y-axis and x = 0
3.) using the formula y = mx + b, m being the slope and b being the y-intercept, you take the information above and plug it in to get y = 2x + 1
i hope this helped!
HELP HELP HELP
HELP HELP HELP
HELP HELP HELP
(DIVISION) NO SYNTHETIC DIVISION NOR FACTORING
STEP BY STEP
Answer:
[tex]2x - 3 + \frac{1}{x-1}[/tex]
Step-by-step explanation:
Attachment :)
We use long division
slope:
y-intercept:
HELP PLS
20 points
Answer:
slope = - 1 , y- intercept = 2
Step-by-step explanation:
calculate the slope m using the slope formula
m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
with ( x₁, y₁ ) = (0, 2) and (x₂, y₂ ) = (2, 0) ← 2 points on the line
m = [tex]\frac{0-2}{2-0}[/tex] = [tex]\frac{-2}{2}[/tex] = - 1
the y- intercept is the point on the y- axis where the line crosses , so
y- intercept = 2
Answer:
Slope: -1
Y-intercept: 2
Step-by-step explanation:
The line crosses the y-axis at 2. and the slope would be -1 since going down by one unit, and the line going down would mean that it is negative.
I hope it helps! Have a great day!
bren~
how do you solve (a²-28)÷(a-5)
Answer:
Divide the highest order term in the dividend a2a2 by the highest order term in divisor aa.Step-by-step explanation:
<3I dont understand:
let CAN be a right triangle with angle C being right angle, side c is 26 cm and side n is 6 cm determine the perimeter of the triangle CAN correct to one decimal enter numerical value only
maybe is my bad english but I dont get it someone can please help?
Using the Pythagorean Theorem, the perimeter of the triangle is of 58.7 cm.
What is the Pythagorean Theorem?The Pythagorean Theorem relates the length of the legs [tex]l_1[/tex] and [tex]l_2[/tex] of a right triangle with the length of the hypotenuse h, according to the following equation:
[tex]h^2 = l_1^2 + l_2^2[/tex]
From your description, I will suppose that the legs are of 6 cm and 26 cm, hence the hypotenuse is:
[tex]h^2 = 6^2 + 26^2[/tex]
[tex]h = \sqrt{6^2 + 26^2}[/tex]
h = 26.7.
The perimeter is the sum of the lengths of the three sides, hence:
P = 26.7 + 26 + 6 = 58.7 cm.
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Some articles were bought at 6 for rupees 5 and sold at 5 for rupees 6. find the gain percent
44% is the answer
6 Article price at buying =5 Rs
1 Article price at buying = 5/6...(i)
5 Articles sold at Rs. 6
1 Articles cost at sold = 6/5 ....(ii)
% Gain=((6/5 - 5/6)/ 5/6) * 100
= 11/25 * 100 = 44%
Profit is a general increase in an asset or the value of an asset. If the item's current price is higher than the original purchase price, you will make a profit. For accounting and tax purposes, profits can be categorized in several ways: B. Gross profit and net profit, or realized profit and unrealized (paper) profit.
The definition of victory is profit, benefit, or increase. An example of profit is a 5% increase in income over the past year. An example of a win is a 5 point lead over another team.
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Find the equation of the linear function represented by the table below in slope-intercept form.
Answer:
y=-2x+3
Step-by-step explanation:
Use the slope formula
[tex]m=\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
According to the table, x=1 and y=1. There is another point x=2 and y=-1.
We can say that [tex]x_{1}=1[/tex] and [tex]y_{1} =1[/tex]. We can also say that [tex]x_{2}=2[/tex] and [tex]y_{2}=-1[/tex]. Plugging this into the equation we get:
[tex]m=\frac{-1-1}{2-1}=-2[/tex].
This means that the slope of your equation is -1/2
Point intercept form is y=mx+b
So plugging the slope in we have [tex]y=2x+b[/tex]
We can plug in one of the values from the value table for x and y, but to keep things simple, I’m going to just use x=1, and y=1, the first values on the table. Plugging these into the equation we get:
[tex]1=-2(1)+b\\ 1=-2 +b\\b=1+2 \\b=3[/tex]
So our equation is [tex]y=-2x+3[/tex]
The graph shows the speed of a cart over a period of several minutes. At approximately what time did the speed of the cart begin to decrease?
The speed of the cart begin to decrease at approximately: 3.5 minutes.
What is a Time-Speed Graph?A typical time-speed graph shows the time on the x-axis and the speed on the y-axis. When the slope is rising, it means the speed is increasing. When the slope is horizontal, it means the speed is constant. When the slope falls or decline, it means speed is decreasing.
We are given a time-speed graph that shows the speed of a cart over a period of several minutes. Time in minutes is on the x-axis (horizontal axis) while speed in miles per hour is on the y-axis (vertical axis).
From the graph given, the speed of the cart started increasing up to about 3 minutes, since the line slopes upwards. Then, between 3 and 4 minutes, we have a horizontal line which starts to decline at approximately 3.5 minutes.
Therefore, we can conclude that the speed of the cart begin to decrease at approximately: 3.5 minutes.
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Which equation is equivalent to 3[x + 3(4x – 5)] = 15x – 24?
15x – 15 = 15x – 24
15x – 5 = 15x – 24
39x – 45 = 15x – 24
39x – 15 = 15x – 24
The equivalent equation of the given equation is 39x - 45 = 15x - 24.
Hence, option C) is the correct answer.
What is the equivalent of the given equation?Given the equation; 3[x + 3(4x – 5)] = 15x – 24
We expand and collect like terms
3[x + 3(4x – 5)] = 15x – 24
3( x + 12x - 15 ) = 15x - 24
3x + 36x - 45 = 15x - 24
We add the like terms on both sides
39x - 45 = 15x - 24
Therefore the equivalent equation of the given equation is 39x - 45 = 15x - 24.
Hence, option C) is the correct answer.
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Select the correct vectors.
A ship moves through the water at 30 miles/hour at an angle of 30° south of east. The water is moving 5 miles/hour at an angle of 20° east of north. Identify the ship's vector, the water current's vector, and the vector representing the ship's actual motion.
Please select the correct answers from the image, thanks and godbless!
In the case above, the correct vectors are:
<25.98, -15>. <1.71, 4.7>.What is the ship vector about?The solution for the Ship's vector are:
Note that the Horizontal aspect = 30 cos 30
= 25.98.
For the Vertical aspect = 30 sin(-30)
= -15.
Hence it will be <25.98, -15>.
In regards to the current's vector:
The Horizontal aspect= 5 sin 20
= 1.71.
The Vertical aspect = 5 cos 20
= 4.7.
Hence it will be <1.71, 4.7>.
Therefore, In the case above, the correct vectors are:
<25.98, -15>. <1.71, 4.7>.Learn more about vectors from
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Of the water lilies in the pond, 43% of them are yellow. The others are white.
A frog randomly jumps onto a lily.
round to the nearest hundredth place
Find the probability of the complement of the frog landing on a yellow lily.
Answer:
The probability of the frog landing on a complement of yellow lily is 0.57
Step-by-step explanation:
PLANTS Of the water lilies in the pond, 43% are yellow. The others are white.
It is given that 43% of plants are lilies it means the probability of a frog randomly jumping onto a lily is:
the probability of the frog landing on a complement of yellow lily is 0.57
WILL MAKE BRAINLIEST!! What type of angle is angle A?
Answer:
Step-by-step explanation:
Obtuse, greater than 90 degrees
Answer:
It should be an obtuse angle
Step-by-step explanation:
Well, it just is
Find the value of X. Round to the nearest tenth HELP ASAP!!!!
Answer: 22.5
Step-by-step explanation:
[tex]\sin 64^{\circ}=\frac{x}{25}\\\\x=25 \sin 64^{\circ}\\\\x \approx 22.5[/tex]
c LM on circle O has a measure of 40°.
Circle O is shown. Line segments L O and M O are radii. Sector L O M is shaded.
Which statements are true? Check all that apply.
The central angle measure created by the shaded region is 40°.
The central angle measure created by the shaded region is 20°.
The ratio of the measure of ∠LOM to the measure of the whole circle is One-ninth.
Circle O can be divided into a total of 9 sectors equal in area to sector LOM.
Circle O can be divided into a total of 10 sectors equal in area to sector LOM.
The statements that are true are;
The central angle measure created by the shaded region is 40°.
The ratio of the measure of ∠LOM to the measure of the whole circle is One-ninth.
Circle O can be divided into a total of 9 sectors equal in area to sector LOM.
What is the central angle?A central angle can be defined as an angle whose vertex is the center O of a circle and whose radii intersecting the circle in two distinct points
Note that;
The two radii are L O and MO
The measure of the angle is 40°
The sector is LOM
From the information given, it can be deduced that the 2 radii that forms the central angle is depicted by the shaded region which is said to have an angle of 40°. The sum of Circle O is one that can be shared into 9 equal sectors and that can be also equal in area to sector that is marked LOM
It can also be deduced that the circle can be divided into 9 equal sectors.
Thus, the statements that are true are;
The central angle measure created by the shaded region is 40°.
The ratio of the measure of ∠LOM to the measure of the whole circle is One-ninth.
Circle O can be divided into a total of 9 sectors equal in area to sector LOM.
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Answer: A,C,D
Step-by-step explanation:
What is the slope of a line that passes through the point (3,-4) and has a y-intercept of -5
Answer:
1/3 = slope
Step-by-step explanation:
Start with
y = mx + b
b = -5 given , now sub in the given point to calculate m =slope
-4 = m(3) - 5 add 5 to both sides
1 = m(3) divide through by 3
1/3 = m = slope
What is the perimeter of the trapezoid
The perimeter of the trapezoid is 18 lines
How to determine the perimeterFormula for perimeter of a trapezoid is given as;
Perimeter = a + b+ c + d
a = 1
b = 4
c = 4
d = 9
Perimeter = 1 + 4 + 4 + 9
Perimeter = 18
Thus, the perimeter of the trapezoid is 18 lines
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A sales manager for an advertising agency believes that there is a relationship between the number of contacts that a salesperson makes and the amount of sales dollars earned. What is the dependent variable
Using variable concepts, of dependent and independent variables, it is found that the dependent variable is the amount of sales dollars earned.
What is the relation between a function and the dependent and independent variables?A function has the following format: y = f(x).In which each value of y is a function of one value of x, and thus, x is the independent variable and y is the dependent variable.That is, the input of the function is the independent variable and the output is the dependent variable.In this problem, the manager believes that the amount of sales dollars earned is a function of the number of contacts that the salesperson makes, hence:
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Select the equivalent expression. (Please Help!!!)
(a/b^3)^4 = ?
Choose 1 answer:
A. a^8/b^8
B. a^4/b^12
C. a/b^81
D. a^5/b^7
Answer: B. a^4/b^12
Step-by-step explanation:
[tex]\large\displaystyle\text{$\begin{gathered}\sf \left(\frac{a}{b^{3} }\right)^{4} \end{gathered}$}[/tex]
Use the rules of exponents to simplify the expression.
[tex]\large\displaystyle\text{$\begin{gathered}\sf \left(\frac{a^{1} }{b^{3}}\right)^{4} \end{gathered}$}[/tex]To raise the quotient of two numbers to a power, raise each power to the power and then divide it.
[tex]\large\displaystyle\text{$\begin{gathered}\sf \frac{(a^{1})^{4} }{(b^{3})^{4} } \end{gathered}$}[/tex]To raise a power to another power, multiply the exponents.
[tex]\large\displaystyle\text{$\begin{gathered}\sf \frac{a^{4} }{b^{3\times4} } \end{gathered}$}[/tex]Multiply 3 by 4.
[tex]\boxed{\large\displaystyle\text{$\begin{gathered}\sf \frac{a^{4} }{b^{12} } \end{gathered}$}}[/tex]We conclude that: the correct option is "B".
A company produces light bulbs each week. Written in scientific notation, which is the best estimate of how many light bulbs the company will make in weeks
The total number of light bulbs the company will make in weeks 9.36 × 10² is 4.27 × [tex]10^{6}[/tex] bulbs.
What is productivity?Work completed in a certain length of time is referred to as productivity (power). One hour, one minute, or one second makes to a unit.
Calculation for the total number of bulbs produced in given week;
Number of bulbs produced per week = 4.56 × 10³
Now, estimate how many bulbs will be produced in 9.36 × 10² weeks.
To estimate how many light bulbs were created in the allotted period, we will simply multiply both values.
Number of light bulbs the company will make in 9.36 × 10² weeks is;
= 4.56 × 10³ × 9.36 × 10²
= 4.27 × [tex]10^{6}[/tex]
Therefore, the total number of bulbs produced in weeks 9.36 × 10² is 4.27 × [tex]10^{6}[/tex] bulbs.
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The complete question is -
A company produces 4.56*10^3 light bulbs each week. Written in scientific notation, which is the best estimate of how many light bulbs the company will make in 9.36*10^2 weeks?
Convert each rate using dimensional analysis. round to the nearest tenth, if necessary. 4.5 qt / min = ____ gal / h a. 67.5 c. 56.3 b. 33.8 d. 65
Option a is correct.
What is dimension analysis?
Dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities and units of measure and tracking these dimensions as calculations or comparisons are performed.
We can convert qt/min to gal/h as shown below:
1 qt = 0.25 gallon
1 min = 1/60 hour
4.5 qt/min = (4.5*0.25)/ (1/60) gal/h
= 4.5*0.25*60 gal/h
= 67.5 gal/h
Hence option a is correct.
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In geometry, Heron's formula (sometimes called Hero's formula), named after Hero of Alexandria, gives the area of a triangle by requiring no arbitrary choice of side as base or vertex as origin, contrary to other formulas for the area of a triangle, such as half the base times the height. Use Heron's formula to find the area, in square meters, of ΔABC.
a=7m, b=5m, c=6m
Answer: S=6√6 m².
Step-by-step explanation:
Heron's formula
[tex]\displaystyle\\\boxed {S=\sqrt{p*(p-a)(p-b)(p-c)} }[/tex],
a, b, c - sides of a triangle,
[tex]\displaystyle\\p=\frac{a+b+c}{2}[/tex]
S - triangle area.
a=7 m, b=5 m, c=6 m.
[tex]p=\frac{7+5+6}{2} \\p=\frac{18}{2} \\p=9 (m).[/tex]
Hence,
[tex]S=\sqrt{9*(9-7)*(9-5)*(9-6)}\\S=3*\sqrt{2*4*3}\\S=3*2*\sqrt{2*3} \\S=6\sqrt{6} \ (m^2).[/tex]
please help me 20 points
Add up the number of students within the random sample data:
32+26+28+22 = 108
We can then use proportional fractions / cross-multiply to answer the question.
In a sample size of 108 students, 32 of them preferred Movie A.
There are 900 students in total.
[tex]\frac{32}{108} =\frac{x}{900}[/tex]
32*900 = 28800
108*x = 108x
108x = 28800
x = [tex]266\frac{2}{3}[/tex] or [tex]\frac{800}{3}[/tex]
In decimal form that is approximately 266.667 and round that to the nearest whole number, you get 267 students who would prefer Movie A.
What is the value of the rational expression
2x- 1
x+5
when x = 0?
Answer:
5
Step-by-step explanation
becasue 2x0=0 and 1x0=0
and when you add 5 to 0 you get five
What is the simplified form of √100x³6?
O 10x¹8
O 10x6
50x18
O 50x6
Question 9 of 10
What is the solution to this equation?
x+12= -5
O A. x = 17
O B. x= -7
O C. x = 7
O D. x= -17
Answer:its b
Step-by-step explanation:
trust me its b
find the volume of a cuboid of length 8cm, width 9cm and height 4cm
Answer:
288 cm³
Step-by-step explanation:
cuboid formular: lengtg×width×height
Answer:
V = 288 cm³
Step-by-step explanation:
the volume (V) of a cuboid is calculated as
V = length × width × height
= 8 × 9 × 4
= 72 × 4
= 288 cm³
8.
P₁V₁ = P₂V₂
T₁ T₂
Rewrite this expression to find T₂.
Answer:
[tex]T_{2} = \frac{P_{2} V_{2} T_{1}} {P_{1} V_{1}}[/tex]
Step-by-step explanation:
The Combined Gas Law expression is:
P₁V₁ / T₁ = P₂V₂ / T₂
To solve for T₂, you need to.....
[tex]\frac{P_{1} V_{1} }{T_{1} } = \frac{P_{2} V_{2} }{T_{2} }[/tex] <------ Original expression
[tex]\frac{T_{2} P_{1} V_{1} }{T_{1} } = {P_{2} V_{2} }[/tex] <----- Multiply both sides by T₂
[tex]{T_{2} P_{1} V_{1} } = {P_{2} V_{2} T_{1}}[/tex] <----- Multiply both sides by T₁
[tex]T_{2} = \frac{P_{2} V_{2} T_{1}} {P_{1} V_{1}}[/tex] <----- Divide both sides by P₁V₁
Solve the equation.
1. for parentheses:
distribute
4-2(x+7) =
3(x+5)
2. if necessary:
combine terms
3. apply properties:
add subtract
multiply
divide
4. to start over:
reset the
The value of equation 4-2(x+7)=3(x+5) is that x=-5/7.
Given equation 4-2(x+7)=3(x+5)
We are required to solve the equation for the value of variable x.
Equation is relationship between two or more variables expressed in equal to form.Equation of two variables looks like ax+by=c. It can be a linear equation,quadratic equation, cubic equation. It can be solved for finding the value of variables.
The given equation is 4-2(9x+2)=3(x+5)
It is a linear equation.
We have to first remove the brackets by multiplying the numbers to inside numbers.
4-18x-4=3x+15
Now collect variable x at one end and constant at other end.
-18x-3x=15-4+4
-21x=15
x=-15/21
x=-5/7
Hence the solution of equation 4-2(x+2)=3(x+5) is that x=-5/7.
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Twenty-three teenagers attended a church camp. This was eleven less than twice the number that attended last year. How many attended last year?
Answer: 17 campers attended last year
Step-by-step explanation:
Let x be the number that attended last year.
The number of campers this year (23) is equal to 2 times the number of last year (x), minus 11.
23 = 2x – 11
23 + 11 = 2x
34 = 2x
34/2 = x
17 = x