Answers
By the alternating series test, this series converges: for [tex]k\in\Bbb N[/tex],
[tex]\dfrac{k^5+1}{k^6+11}[/tex]
is a positive, decreasing sequence that converges to 0.
However,
[tex]\displaystyle \sum_{k=2}^\infty \left| (-1)^{k+1} \frac{k^5+1}{k^6+11} \right| = \sum_{k=2}^\infty \frac{k^5+1}{k^6+11} \approx \sum_{k=2}^\infty \frac1k[/tex]
is a divergent series by comparison to the harmonic series.
So the given series is conditionally convergent.
The given series is conditionally convergent. This can be obtained by using alternating series test first and then comparing the series to the harmonic series.
Determine if diverges, converges, or converges conditionally:Initially we need to know what Absolute convergence and Conditional convergence,
If [tex]\sum|a_{n} |[/tex] → converges, and [tex]\sum a_{n}[/tex] → converges, then the series is Absolute convergence
If [tex]\sum|a_{n} |[/tex] → diverges, and [tex]\sum a_{n}[/tex] → converges, then the series is Conditional convergence
First use alternating series test,
[tex]\lim_{k \to \infty} \frac{k^{5} +1}{k^{6}+11 }[/tex] = [tex]\lim_{n \to \infty} \frac{5}{6k}[/tex] = 0,
The series is a positive, decreasing sequence that converges to 0.
Next by comparing the series to harmonic series,
[tex]\sum^{\infty} _{k=2}|(-1)^{k+1} \frac{k^{5} +1}{k^{6}+11 }|=\sum^{\infty} _{k=2}\frac{k^{5} +1}{k^{6}+11 }[/tex] ≈ [tex]\sum^{\infty} _{k=2}\frac{1}{k}[/tex] = 0
This implies that the series is divergent by comparison to the harmonic series.
First we got that the series is converging and then we got the series is divergent. Therefore the series is conditionally convergent.
[tex]\sum|a_{n} |[/tex] → diverges, and [tex]\sum a_{n}[/tex] → converges, then the series is Conditional convergence.
Hence the given series is conditionally convergent.
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Related Questions
Identify the slope of a line perpendicular to the given line Y=2x-4
Answers
[tex]m = \frac{ - 1}{a} = \frac{ - 1}{2} \\ y = \frac{ - 1}{2} x + b[/tex]
The sum ∫2−2() ∫52()−∫−1−2() can be written as a single integral in the form ∫() determine and
Answers
We have
[tex]\displaystyle \int_{-2}^2 f(x) \, dx - \int_{-2}^{-1} f(x) \, dx = \int_{-1}^2 f(x) \, dx[/tex]
so that
[tex]\displaystyle \int_{-2}^2 f(x) \, dx + \int_2^5 f(x) \, dx - \int_{-2}^{-1} f(x) \, dx \\\\ ~~~~~~~~~~~~ = \int_{-1}^2 f(x) \, dx + \int_2^5 f(x) \, dx \\\\ ~~~~~~~~~~~~ = \boxed{\int_{-1}^5 f(x) \, dx}[/tex]
find PQ if possible
Answers
Answer: 11 units
Step-by-step explanation:
We can say that [tex]\triangle TSQ\sim\triangle PSR[/tex] by AA Similarity Postulate. This is since [tex]\angle T\cong\angle P[/tex] and both ∠TSQ and ∠RSP are right angles, making them congruent.
Similar triangles have a property that corresponding sides are proportional. Hence, we can say that
[tex]\frac{ST}{PS}=\frac{SQ}{SR}\\\frac{15}{PS}=\frac{9}{12}\\\frac{15}{PS}=\frac{3}{4}\\60=3*PS\\PS=20[/tex]
We also know that PS is the combined length of PQ and QS. Since we know that QS is 9, let's substitute PQ + 9 in and solve.
[tex]PQ+9=20\\PQ=11[/tex]
Select the correct answer.
What is the solution to the equation?
2(x + 7) = 8
OA. -15 and 1
OB. -15
OC. 1
OD.
no solution
Answers
Answer:
Option D is correct.
No solution
Step-by-step explanation:
2(x+7)=8
Apply 2 into bracket:
2x+14=8
2x=8-14
2x=-6
Divide both sides by 2:
x=-3
hence here is no option for x=-3, so answer is no solution.
Four interior angles of a pentagon measure 156°, 72°, 98°, and 87°. What is the measure of the final interior angle?
108°
127°
150°
487°
Answers
Answer:
127
Step-by-step explanation:
Answer:
b
Step-by-step explanation:
40 points answer ASAP PLEASE!
For what values of x does f(x) = 0?
A.−4, 0, 2
B.−2, 0, 4
C.−5, 0, 17
D.−17, 0, 5
Answers
I think it is 2, 0, 4 but it would be very helpful if you put in an image of the graph.
Have a great day :D
answer????? with steps
Answers
=600x5
=3000
i need help with this ;n;
Answers
I dont know this, please can someone help
Answers
The table of values for y=1/2x-1 is completed as follows:
x y
-2 -2
-1 -3/2
0 -1
1 -1/2
2 0
3 1/2
Exactly one value from the set of second components of the ordered pair is connected with each value from the set of first components of the ordered pairs in a relation known as a function.
Given function: y=1/2x-1
For x = -2
Value of y = -2
For x = -1
Value of y = 1/2(-1)-1 = -3/2
For x = 0
Value of y = 1/2(0)-1 =-1
For x = 1
Value of y = 1/2(1)-1 = -1/2
For x = 2
Value of y = 0
For x = 3
Value of y = 1/2(3)-1 = 1/2
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How many integers between $1$ and $200$ are multiples of both $3$ and $5$ but not of either $4$ or $7$
Answers
There are only 13 multiples of both 3 and 5, since
[tex]200 = 3\cdot5\cdot13 + 5[/tex]
From these integers, we eliminate any that are also divisible by 4 or 7.
[tex]3\cdot5\cdot4 = 60 \implies \{60,120,180\}[/tex]
[tex]3\cdot5\cdot7 = 105 \implies \{105\}[/tex]
so there are 13 - 4 = 9 such integers.
On a coordinates plane, a line passes through ( - 2 , 3 ) and ( 2 , 6 ). Which of the followings lies on the same line.
a. ( - 2 , 12 )
b. ( 6 , 12 )
c. ( - 6 , 0 )
d. ( - 6 , 6 )
Answers
By direct comparison and definition of line segment we notice that the point (x, y) = (- 6, 0) lies on the line segment AB as each AP is a multiple of former. (Correct choice: C)
What point lies on a line segment?
According to linear algebra, a point lies in a line segment if its vector is a multiple of the vector that generates the line segment itself, that is:
AB = k · AP (1)
The vector that generates the line segment is:
AB = (2, 6) - (- 2, 3)
AB = (4, 3)
And the vectors related to each point are:
Case A
AP = (- 2, 12) - (- 2, 3)
AP = (0, 9)
Case B
AP = (6, 12) - (- 2, 3)
AP = (8, 9)
Case C
AP = (- 6, 0) - (- 2, 3)
AP = (- 4, - 3)
Case D
AP = (- 6, 6) - (- 2, 3)
AP = (- 4, 3)
By direct comparison and definition of line segment we notice that the point (x, y) = (- 6, 0) lies on the line segment AB as each AP is a multiple of former. (Correct choice: C)
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PLSSS
C= [tex]\frac{12^{3} . 6^{5} }{9^{4} . 2^{10}}[/tex]
Answers
Answer:
2
Step-by-step explanation:
The trick here is to reduce each of the bases to lowest form first
[tex]12 = 3.4 = 3 .2.2 = 3.2^{2}\\12^{3} = (3.2^{2} )^{3} = 3^{3}2^{6} \\6 = 3.2\\6^{5} = (3.2)^{5} = 3^{5} .2^{5} \\\\Numerator = 3^32^63^52^5 = 3^82^{11}\\Denominator computation\\9 = 3^2\\9^{4} = (3^{2} )^4 = 3^{8} \\Denominator = 3^82^{10} \\\\\frac{Numerator}{Denominator } =\frac{{3^8}{2^{11} }}{3^82^{10}} = \frac{2^{11}} {2^{10}} = 2^1 = 2[/tex]
What is the area of the sector of the circle below, if the radius is 5 m. and the central angle < AOB measures 88 °. (round answer to the nearest tenth)
Answers
Answer:
b 19.2
Step-by-step explanation:
a = [tex]\pi[/tex][tex]r^{2}[/tex] for a circle. We do not want to find the area for a whole circle. We only want to find the area for part of a circle. a hole circle is 360 degrees.
a = [tex]\frac{88}{360}[/tex][tex]\pi[/tex][tex]r^{2}[/tex]
a = [tex]\frac{88}{360}[/tex][tex]\pi[/tex]([tex]5^{2}[/tex])
a = 19.2 rounded.
The area of the sector of circle is b. 19.20 square meter.
What is the area of sector of circle?The space enclosed by the sector of circle is called area of the sector of circle.Mathematically,
Area of the sector of circle, A = θ/360πr²
where θ is the angle of the arc and r is the radius of the circle.
Now it is given that,
radius of the circle, r = 5m
Angle of arc, θ = 88°
Therefore, area ofsector of circle A = θ/360πr²
Put the values,
A = 88/360 π 5²
Solving the equation we get
A = 19.20 square meter.
Hence,the area of the sector of circle is b. 19.20 square meter.
So the correct answer is b.) 19.20 square meter.
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Find the area of the base of a drum if it needs 88 m long rope to tie twice it around.
solve with full solutions!
Answers
Answer:
[tex]154 $\ m^2[/tex]
Step-by-step explanation:
If 88 m of long rope is wrapped twice around the base of the drum, the circumference of the drum is 44m. We can use this information to find the radius of the base.
Find the RadiusThe equation for circumference is [tex]C=2\pi r[/tex]. We can substitute the value we found for circumference into the equation:
[tex]44=2\pi r[/tex]
Divide both sides by 2
[tex]22=\pi r[/tex]
Divide both sides by π
[tex]r=\frac{22}{\pi}[/tex]
Use the Radius to find the AreaThe area of a circle is [tex]A=\pi r^2[/tex]. We can substitute the value we found for the radius into the equation:
[tex]A=\pi (\frac{22}{\pi})^2\\[/tex]
[tex]A=\pi (\frac{22}{\pi})(\frac{22}{\pi})[/tex]
[tex]A=22*\frac{22}{\pi}[/tex]
[tex]A=\frac{484}{\pi}[/tex]
[tex]A\approx154[/tex]
if the earth is 12.5 million years old, how old is the earth in hours?
Answers
Answer:
1.09575e+11
Step-by-step explanation:
365 days per year
12,500,000 years
12,500,00*365*24=
1.095 x 10^11
Given u = ⟨3, 2⟩, v = ⟨1, 5⟩, and the angle between the vectors is 45°, what is u · v?
Answers
We need to use a technique called dot multiplication. In this technique, we multiply the magnitude of two given vectors by the value of the angle between them.
In the question, the value of the angle between two vectors [tex]cos(45^0)[/tex] and the coordinates of the two vectors are given. Let's multiply them by the dot product to get the result.
Formula: [tex]u.v=|u||v|cos(a)[/tex]
[tex]u=3i+2j[/tex]
[tex]v=i+5j[/tex]
[tex]cos(a)=\frac{1}{\sqrt{2}}[/tex]
To find the magnitude of a vector, we square the coordinates, add them up and take the square root of the result.
[tex]Magnitude=\sqrt{x^2+y^2+z^2+...}[/tex]Result: [tex]u.v=|\sqrt{(3^2)+(2^2)}||\sqrt{(1^2)+(5^2)}|.\frac{1}{\sqrt{2}}=(13\sqrt{2})\frac{\sqrt{2}}{2} =13[/tex]
Scalcet8 4. 2. 26. suppose that 2 ≤ f '(x) ≤ 4 for all values of x. what are the minimum and maximum possible values of f(4) − f(1)? ≤ f(4) − f(1) ≤ show my work (optional)
Answers
The minimum value of f(4) - f(1) is 6.
The maximum value of f(4) - f(1) is 12.
In the question, we are given that, 2 ≤ f'(x) ≤ 4 for all values of x.
Taking the given inequality as (i).
We are asked to find the minimum and maximum possible values of f(4) - f(1).
We multiply (i) by dx throughout, to get:
4dx ≤ f'(x)dx ≤ 5dx.
To find this, we integrate (i) in the definite interval [4, 1] with respect to dx, to get:
[tex]\int_{1}^{4}2dx \leq \int_{1}^{4}f'(x)dx \leq \int_{1}^{4}4dx\\\Rightarrow [2x]_{1}^{4} \leq [f(x)]_{1}^{4} \leq [4x]_{1}^{4}\\\Rightarrow 2*4 - 2*1 \leq f(4)-f(1) \leq 4*4 - 4*1\\\Rightarrow 6 \leq f(4) -f(1) \leq 12[/tex]
Thus, the minimum value of f(4) - f(1) is 6.
The maximum value of f(4) - f(1) is 12.
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what is the equivalent expression (3*5)^6=?
Answers
Answer:
15^6
Step-by-step explanation:
According to BODMAS
where;
B - Bracket ()
O - Off (*)
D - Division (/)
M - Multiplication (*)
A - Addition (-)
S - Subtraction (+)
Bracket should be solved first
therefore;
(3*5)^6
15^6 = 11390625
Which can be approximately in standard form written as 1.1*10^7
The geometry of the hybrid orbitals about a central atom with sp3d hybridization is:________
Answers
For sp3d hybridized central atoms, the only possible molecular geometry is trigonal bipyramidal.
Hybridisation (or hybridization) is a process of mathematically combining two or more atomic orbitals from the same atom to form an entirely new orbital different from its components and hence being called as a hybrid orbital.
In this case, if all the bonds are in place, the shape is also trigonal bipyramidal.
In chemistry, a trigonal bipyramid formation is a molecular geometry with one atom at the center and 5 more atoms at the corners of a triangular bipyramid.
Bond angle(s): 90°, 120°
Coordination number: 5
Point group: D3h
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WILL GIVE CROWN
What is the remainder of 2x^2+7x-39/x-7
show work
Answers
On dividing we get 69/2 or 34.5 as remainder.
In algebra, the factor theorem is a theorem linking factors and zeros of a polynomial. It is a special case of the polynomial remainder theorem. The factor theorem states that a polynomial f(x) has a factor if and only if f=0.
Long division is best suited for such expressions:-
On applying long division and factor theorem we get
2x²+7x-39 = (2x - 1/2)(x-7) + 69/2
Thus on dividing we get 69/2 or 34.5 as remainder.
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what is 5/8 divided by 1/2?
Answers
Answer:
5/4
Step-by-step explanation:
(5/8)(2/1)
(Reciprocal of 1/2 is 2/1 as shown above)
(5)(2)/(8)(1)=10/8
10/8=5/4
Saul walks dogs to earn extra money. He
earned $220 last week by walking 16 dogs.
Write and solve an equation to determine
how much he charges to walk each dog
each week.
Answers
Answer:
£13.75
Step-by-step explanation:
→ Call the charge x
x
→ Multiply by 16
16x
→ Equate to total
16x = 220
→ Divide both sides by 16
x = £13.75
help needed with this one
Answers
Based on the given parameters, the length of the missing length in the triangle is 12 units
How to determine the length of the missing side?The given triangle is an scalene triangle
Scalene triangles are triangles that have unequal three sides
The missing side is x, and the value of x is calculated using the following equivalent ratios
We have
4 : 8 = 6 : x
Express the above equation as a fraction
4/8 = 6/x
Evaluate the quotient
0.5 = 6/x
Solve for x
x = 6/0.5
Evaluate the quotient
x = 12
Hence, the length of the missing length is 12 units
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Jamie is 6 years older than tyrion now. in 2 years, jamie will be one year less than twice tyrion's current age. what is jamie's current age?
Answers
Answer:
Jamie is currently 15.
Step-by-step explanation:
Define the singular matrix.
Answers
Answer:
A square matrix whose determinant is equal to zero is called singular matrix.
[tex]\huge \mathbb{ \underline{ANSWER:}}[/tex]
[tex]\leadsto[/tex] So the square matrix that has its det zero and its inverse is not defined is called a singular matrix.
[tex]\bold{Example:}[/tex]
[tex]\begin{bmatrix} \sf{3} & \sf{6} \\ \sf{2} & \sf{4}\end{bmatrix} \\ \\ \sf and \\ \begin {bmatrix} \sf{ 1} & \sf2 & { \sf2} \\ { \sf1} & { \sf2} & \sf{2} \\ \sf{3} & \sf {2} & \sf{1}\end{bmatrix}[/tex]
[tex]\huge \mathbb{ \underline{EXPLANATION:}}[/tex]
A singular matrix is a square matrix if its determinant is O. i.e., a square matrix A is singular if and only if det A = 0. The formula to find the inverse of a matrix is
▪ [tex] \bold{ {A}^{1} = \dfrac{1 \: }{det \:A } adj[A] }[/tex]
So if det A = 0 then the inverse of A will be not defined.
M angle JKL=
Help me please! Asap thanks so much :)
Answers
The measure of the angle <JKL is 56 degrees
Tangent secant theorem of a circleThe theorem states that if a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.
Given the following parameters
m<IL = 112 degrees
Since the measure of the vertex is half the measure of its intercepted arc, hence;
<JKl = 1/2(m<IL)
Substitute the given parameters into the formula to have;
<JKl = 1/2(112)
<JKL = 56 degrees
Hence the measure of the angle <JKL is 56 degrees
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find the value of x and y that satisfies both the equaties x/y= 2/3 and y/24=3/4
Answers
Answer:[tex]\Large\boxed{x=12~;~y=18}[/tex]
Step-by-step explanation:
Given a system of equations
[tex]1)~\frac{x}{y} =\frac{2}{3}[/tex]
[tex]2)~\frac{y}{24} =\frac{3}{4}[/tex]
Multiply 24 on both sides of 2) equation to isolate y-value
[tex]\frac{y}{24}\times24 =\frac{3}{4}\times24[/tex]
[tex]\Large\boxed{y=18}[/tex]
Substitute the y-value into the 1) equation
[tex]\frac{x}{18} =\frac{2}{3}[/tex]
Multiply 18 on both sides of 1) equation to isolate y-value
[tex]\frac{x}{18}\times18 =\frac{2}{3}\times18[/tex]
[tex]\Large\boxed{x=12}[/tex]
Hope this helps!! :)
Please let me know if you have any questions
What are the points if the slope is 4 and the point is (1,-1)
Answers
The points if the slope is 4 and the point is (1,-1) are (1,-1) and (2, 3)
What are linear equations?Linear equations are equations that have constant average rates of change, slope or gradient
How to determine the points?The given parameters are:
Slope = 4
Point = (1, -1)
The slope of 4 means that as x changes by 1, the value of y changes by 4
This means that, we have:
(x + 1, y + 4)
Substitute the known values in the above equation
(1 + 1, -1 + 4)
Evaluate
(2, 3)
Hence, the points if the slope is 4 and the point is (1,-1) are (1,-1) and (2, 3)
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Use the bisection method to find solutions accurate to within 10−2 for x 4 − 2x 3 − 4x 2 4x 4 = 0 on each interval.
(a) [−2, −1].
(b) [0, 2].
(c) [2, 3].
(d) [−1, 0].
Answers
The solution accurate to equation within [tex]10^{-2}[/tex] for [tex]x^{4}-2x^{3} -4x^{2} +4x+4=0[/tex] lies in [0,2].
Given the equation [tex]x^{4}-2x^{3} -4x^{2} +4x+4=0[/tex] and range is [tex]10^{-2}[/tex].
We are required to find the interval in which the solution lies.
The attached table shows the iterations. At each step, the interval containing the root is bisected and the function value at the mid point of the interval is found. The sign of its relative to the signs of the function values at the ends of the interval tell which half interval contains the root. The process is repeated until the interval width is less than [tex]10^{-2}[/tex].
Interval:[0,2], signs [+,-],mid point:1, sign at midpoint +.
[1,2] 3/2
[1,3/2] 5/4
The rest is in the attachment. The listed table values are the successive interval mid points.
The final midpoint is 181/128=1.411406.
This solution is within 0.0002 of the actual root.
Hence the solution accurate to equation within [tex]10^{-2}[/tex] for [tex]x^{4}-2x^{3} -4x^{2} +4x+4=0[/tex] lies in [0,2].
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Charlie began hiking from her campsite back to her car. Her campsite was located 9 miles from her car. She had walked for 1 hour and was 2 miles from her campsite when she realized she had forgotten her water bottle. She hurried back to her campsite in 0.5 hours, collected her water bottle, and rested for 0.5 hours. Then she began hiking back to her car at a quicker pace. She hiked for 3 more hours before she reached her car. Which graph represents Charlie's distance from her car at different times?
Answers
The Distance - Time graph that represents that represents Charlie's trip is attached accordingly.
What is a Distance - Time Graph?A distance-time graph can be used to illustrate the distance traveled by an item moving in a straight line.
The gradient of the line in a distance-time graph equals the speed of the item. The quicker the item moves, the higher the gradient (and the steeper the line).
What is the use of a Distance - Time Graph?Some of the uses of the distance-time graph are;
The position of the object at a time can be detected using the distance - time graphThe D-T Graph can provide the speed at which a person or an object is traveling.Learn more about Distance - Time Graph:
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