If c is a positive real number and ak=klnc, for what values of c, if any, does the infinite series ∑k=1[infinity]ak converge?

Answers

Answer 1

There is no value of c such that the infinite series converges.

How to analyze the converge of an infinite series

Convergence is a characteristic of infinite series, in which a sum tends to be a given number when n tends to +∞.

In this question we must prove for which values of c the infinite series may converge.

There are different criteria to prove whether a given series converges or not, one of the most used criteria is the ratio criterion, which states that:

[tex]r = \frac{a_{k+1}}{a_{k}}[/tex], where the series converges if and only if r > 1. (1)

If we know that [tex]a_{k} = k\cdot \ln c[/tex], then the rational formula is:

[tex]r = \frac{(k+1)\cdot \ln c}{k\cdot \ln c}[/tex]

[tex]r = 1 + \frac{1}{k}[/tex]

We notice that resulting expression does not depend on c and is greater than 1. Therefore, there is no value of c such that the infinite series converges. [tex]\blacksquare[/tex]

Remark

If [tex]c[/tex] is a positive real number and [tex]a_{k} = k\cdot \ln c[/tex], for what values of [tex]c[/tex], if any, does the infinite series [tex]\sum_{k=1}^{\infty} a_{k}[/tex] converge?

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Answer 2

There is no values of c, for  the infinite series ∑k=1[infinity]ak to be converge when c is a positive real number.

What do you mean by infinite series?

The infinite series is the sum of addition of a sequence in which the number of terms present are infinite.

[tex]a_1+a_2+a_3....+a_n[/tex]

Here, ([tex]a_n[/tex]) is the nth term of the sequence. For infinite series, the ratio criterion can be given as,

[tex]r=\dfrac{a_{k+1}}{a_k}[/tex]

If c is a positive real number and

[tex]a_k=k\ln c[/tex]

Put the value in above shown ratio,

[tex]r=\dfrac{(k+1)\ln c}{k\ln c}\\r=\dfrac{k\ln c}{k\ln c}+\dfrac{\ln c}{k\ln c}\\r=1+\dfrac{1}{k}[/tex]

With the value of r, we can say whether the series is converge or not. This value of r does not depends on c.

Thus,  there is no values of c, for  the infinite series ∑k=1[infinity]ak to be converge when c is a positive real number.

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Related Questions

A fighter plane is descending at 30 m/s. The pilot ejects, and the ejector seat accelerates him upwards at 120 m/s2 for 2 seconds. What is his velocity at the end of the 2 seconds?

Answers

Answer:

210m/s upwards

Explanation:

u= -30m /s

a= 120m/s^2

t=2s

v=u+at

=(-30)+(120x2)

=240-30

=210m/s

amy bought 3 body lotions and 7 body spray for 80 jessica bought 6 body lotion and 3 body spray for 63.75 how much does one body spray cost

Answers

Answer:

Each body spray cost $ 21.25

Explanation:

A student looks at a white light source through spectroscope which views the light through many tiny slits. The student sees a rainbow of colors. Which of these is the best explanation for why the student sees many colors through the slits.

Answers

When the various colors in the spectrum of white light are separated, the student sees a rainbow of colors.

Spectrum of white light

White light is a spectrum of seven wavelengths namely; red, orange, violet, green, blue, indigo, yellow. When white ligt is passed through many tiny slits, these wavelengths become separated.

As a result of the separation of these seven wavelengths that compose white light, the student sees a rainbow of colors.

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How to know if two vectors are linearly independent

Answers

Answer:

Given a set of vectors, you can determine if they are linearly independent by writing the vectors as the columns of the matrix A, and solving Ax = 0.

Explanation:

You may verify if a group of vectors is linearly independent by expressing the vectors as columns of the matrix A and solving Ax = 0.

What is a vector?

A vector is a quantity or phenomena with magnitude and direction that are independent of one another. The phrase also refers to a quantity's mathematical or geometrical representation.

If no vector can be written as a linear combination of the others, a set of vectors is said to be linearly independent.

If no vector in a collection of vectors can be written as a linear combination of the ones mentioned before it, the set is said to be linearly independent.

A nontrivial linear combination yields the zero vector, the vectors are said to be dependent.

You may verify if a group of vectors is linearly independent by expressing the vectors as columns of the matrix A and solving Ax = 0.

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What is the ultimate fate of an isolated white dwarf?.

Answers

Answer:

It will cool down and become a cold black dwarf.

Explanation:

The electron degeneracy pressure slowly overwhelms gravity, and the white dwarf evaporates. As gravity overwhelms the electron degeneracy pressure, it will explode as a nova.

What is the equation for finding the amount of inertia in forces and motion? Explain how you solve that equation.​

Answers

Answer:

Inertia = angular momentum / angular velocity

What is the speed of a wave that has a frequency of 173 Hz and a wavelength of 2. 59 meters? Express your answer to the nearest whole number. M/s.

Answers

The speed of the wave will be v= 448.07 m/s

What is speed of the wave?

The speed of the wave is defined as the distance traveled by the wave with respect to the time.

The frequency is defined as the number of cycles per seconds it is calculated in hertz

The distance between two successive troughs or crests is known as the wavelength. The peak of the wave is the highest point, while the trough is the lowest.

Now frequency = 173 hertz

wavelength = 2.59 m

The velocity will be given by the formula:

[tex]v=\lambda\times f[/tex]

[tex]v=173\times 2.59=488.07 \ \frac{m}{s}[/tex]

Thus the speed of the wave will be v= 448.07 m/s

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write comparison between Electrical Circuit and Magnetic Circuit.

[tex] \\ \\ \\ \\ \\ \\ \\ [/tex]


Thanks¡!¡:)​

Answers

Magnetic circuit, closed path to which a magnetic field, represented as lines of magnetic flux, is confined.

In contrast to an electric circuit through which electric charge flows, nothing actually flows in a magnetic circuit.

Explanation:

Hope it helps you (`▿▿▿▿´ƪ)

If a food has 200 calories and 7 grams of protein, and each gram of protein is 4 calories, what percent of the serving comes from protein?

Answers

Answer: 13%

Explanation: The reason this is 13 is because you can eat carlories and you can also be an amazing person 200 calories and 7 grams

A thin rod of length d on a frictionless surface is pivoted about one end

Answers

The magnitude of the angular momentum of the rod immediately after the collision is given as (pf + pi) * d. (Option A).

What is Angular Momentum?

Angular Momentum may be defined or described as the vector quantity of the rotation of a body, which is arrived at by multiplying its moment of inertia by its angular velocity.

The formula for Angular Momentum is given as:

L = mvr; Where

L = Angular Momentum

m = mass

v = velocity; and

r = radius.

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[tex]\rm( P_f+P_i)d[/tex] is the magnitude of the rod's angular momentum immediately after the contact. Option A is correct.

What is the definition of Angular Momentum?

Angular Momentum may be defined as the vector quantity of a body's rotation, which is calculated by multiplying its moment of inertia by its angular velocity.

The Angular Momentum formula is as follows:

[tex]\rm L=I\omega[/tex]

[tex]\rm L = mvr[/tex]

Where,

L is the angular momentum

m stands for mass.

v stands for velocity, and

r stands for radius.

Hence option A is correct.

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which change of state is shown in the photo

A) Liquid to solid
B) Liquid to gas
C) Gas to liquid
D) Solid and liquid

Answers

The change of state that is shown here is from solid to liquid. The correct option is D.

What is states of matter?

A state of matter is one of the various forms that matter can take in physics. In everyday life, four states of matter are visible: solid, liquid, gas, and plasma.

In most environments, matter can exist in three distinct physical states: solid, liquid, and gas. In extreme environments, other states such as plasma, Bose-Einstein condensate, and neutron stars may exist.

Here in the given image, ice which is a solid is getting transformed into water i.e., liquid, so it can be considered that the phase is transforming from solid to liquid.

Thus, the correct option is D.

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What is the tension of the string if the cylinder has an acceleration of

Answers

In physics, the tension on a suspended object said to be equal to the weight of the object plus or minus the mass of the object times the acceleration acting on it.

~ ISS

If a peregrine falcon dove at its maximum speed for half a mile to catch prey, how many seconds would the dive take? (round your answer to the nearest second. ) correct answer9

Answers

The number of seconds that the dive will take if peregrine falcon dove at its maximum speed for half a mile is 9 seconds.

How to calculate the time?

From the complete information, it was stated that the peregrine falcon can reach speed of up to 200 miles per hour.

Therefore, the number of seconds that the dice would take for half a mile will be calculated thus:

1 hour = 200 miles

3600 seconds = 200 miles

1 second will be: = (200/3600) = 0.056 miles

Therefore, the time will be:

200miles / 1 hour = 0.x

x = 0.0025 hours

Therefore, 0.0025 hours will be:

= 0.0025 × 3600

= 9 seconds.

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a body has a mass 5 kg what does it mean​

Answers

Answer:

If the mass of an object is 5kg it means that the mass is measured on the unit kilogram and this unit is contained 5times the given quantity of the object.

If an ocean wave passes a stationary point every 5 s and has a velocity of 10 m/s, what is the wavelength of the wave

Answers

Answer:

As Per Provided Information

Velocity of wave v is 10m/s

These ocean wave passes a stationary point every 5 s ( It's time period)

First we calculate the frequency of ocean wave .

Using Formulae

[tex] \blue{\boxed{\bf \: \nu = \cfrac{1}{v}}}[/tex]

here

v is the velocity of wave .

[tex]\sf\longrightarrow \nu \: = \cfrac{1}{10} \\ \\ \\ \sf\longrightarrow \nu \: = 0.1Hz[/tex]

Now calculating the wavelength of the wave .

Using Formulae

[tex] \boxed{ \bf \lambda = \cfrac{v}{ \nu}}[/tex]

Substituting the value and we obtain

[tex] \sf \longrightarrow \lambda \: = \cfrac{10}{0.1} \\ \\ \\ \sf \longrightarrow \lambda \: = \cancel\cfrac{10}{0.1} \\ \\ \\ \sf \longrightarrow \lambda \: =100m[/tex]

Therefore,

Wavelength of the wave is 100 metres.

Cars enter a car wash at a mean rate of 33 cars per half an hour. What is the probability that, in any hour, exactly 22 cars will enter the car wash? round your answer to four decimal places

Answers

Answer:Cars enter a car wash at a mean rate of 4 cars per half an hour. 

Explanation:

The probability that, in any hour, exactly 22 cars will enter the car wash will be 0.0920.

What is probability?

Probability means possibility. It deals with the occurrence of a random event. The value of probability can only be from 0 to 1.

Its basic meaning is something is likely to happen. It is the ratio of the favorable event to the total number of events.

The  mean rate of car is,[tex]\rm \lambda = 66 \ car/hour[/tex]

[tex]P(x=K) = \frac{8^k e^{-8}}{k\!}\\\\P(x=22) = \frac{66^{22} e^{-22}}{22\!}\\\\ P(x=22)=\frac{0.000000000000691\times 1.07\times 10^{40}}{22} \\\\\ P(x=22)=0[/tex]

Hence the probability that, in any hour, exactly 22 cars will enter the car wash will be 0

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15. A 0.500-kg mass suspended from a spring oscillates with a period of 1.50 s. How much mass must be added to the object to change the period to 2.00 s?

The answer is 0.389 kg but please show your work.

Answers

Let's look at relationship

[tex]\\ \rm\rightarrowtail T=2\pi\sqrt{\dfrac{m}{k}}[/tex]

[tex]\\ \rm\rightarrowtail T\propto \sqrt{m}[/tex]

Hence

[tex]\\ \rm\rightarrowtail \dfrac{T_1}{T_2}=\sqrt{\dfrac{m1}{m2}}[/tex]

[tex]\\ \rm\rightarrowtail \dfrac{1.5}{2}=\sqrt{\dfrac{0.5}{m2}}[/tex]

[tex]\\ \rm\rightarrowtail 0.75^2=\dfrac{0.5}{m2}[/tex]

[tex]\\ \rm\rightarrowtail m_2=\dfrac{0.5}{0.75^2}[/tex]

[tex]\\ \rm\rightarrowtail m_2=0.889[/tex]

Hence

Mass needs to added =0.889-0.500=0.389kg

Answer

Mass needs to added =0.889-0.500=0.389kg

Explanation:

A particular lightbulb is designed to consume 40 W when operating on a car's 12-V DC electric power. If you supply that bulb with 12-V AC power from a transformer, how much power will it consume

Answers

40V because it will provide the same amount of power.

A photon with wavelength 11. 0 nm is absorbed when an electron in a three-dimensional cubical box makes a transition from the ground state to the second excited state

Answers

For a photon with wavelength 11. 0 nm,  the side length LL of the box is mathematically given as

L=1.414*10^{-27}m

What is the side length LL of the box?

Generally, the equation for the change in energy is mathematically given as

[tex]dE=\frac{hc}{\lambda}[/tex]

Therefore

[tex]\frac{hc}{\lambda}=\frac{6h^2}{8ml}\\\\L=\sqrt{6h\lambda}{8mc}\\\\L=\sqrt{\frac{6(6.626*10^{-34})*11.0*10^{-9}}{8(9.11*10^31)(3*10^8)}}[/tex]

L=1.414*10^{-27}m

In conclusion,  the side length LL of the box is

L=1.414*10^{-27}m

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The side length of the box for a photon of wavelength 11. 0 nm will be [tex]\rm L = 1.414 \times 10^{-27}[/tex] m.

What is wavelength?

The distance between two successive troughs or crests is known as the wavelength. The peak of the wave is the highest point, while the trough is the lowest.

The change in the energy of the photon is found as;

[tex]\rm dE = \frac{hc}{\lambda} \\\\ \frac{hc}{\lambda} =\frac{6h^2}{8ml} \\\\ L= \sqrt{\frac{6h\lambda}{8mc} } \\\\ L = \sqrt{\frac{6 \times 6.626 \times 10^{-34}\times 11 \times 10^{-9}}{8 \times 9.11 \times 10^{31} \times 3 \times10^8} } \\\\ L= 1.414 \times 10^{-27} \ m[/tex]

Hence  the side length of the box for a photon of wavelength 11. 0 nm will be [tex]\rm L = 1.414 \times 10^{-27}[/tex] m.

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Today, you will use the Magnetic Fields Simulation to collect more reliable data about the two
variables that were tested on the eliminated cards: the number of wire coils on the electromagnet
and the distance between the car and launcher.
1. How does the number of wire coils on an electromagnet affect the strength of magnetic force?
Describe how you will set up tests with isolated variables in the Sim to answer this question.
What will you change?
What will you keep the same?
What will you measure?

Answers

The current flowing in the wires will be kept constant while the number of turns of the wires will be changed, then you will measure the strength of the magnetic force.

Strength of the magnetic force

The strength of the magnetic force depends on the number of the turns of the wire and the amount of current flowing in the wires in a given magnetic field.

The relationship between the magnetic force and the number of wire coils on an electromagnet is given as;

F = I  x N

where;

I is the current in the wire N is the number of turns

Thus, the current flowing in the wires will be kept constant while the number of turns of the wires will be changed, then you will measure the strength of the magnetic force.

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Photoelectrons are emitted from the surface of a metal when light shines on it. If the wavelength of the light is decreased

Answers

Answer:

what are your answer choices if you have any?

Explanation:

(my ask question thingy isnt working)

1 point
8. A magnetic field of 30 T is directed perpendicular to the force of a coil
with an area of 0.36 m^2. Calculate the magnetic flux in the coil.

A)0.27 Wb
B)9.3 Wb
C)10.8 Wb
D)83.3 Wb

Answers

Answer:

the answer is. C.

Explanation:

[tex] 0.36 \times 30 = 10.8[/tex]

the solution

At what constant velocity must a spacecraft travel from earth if it is to reach the star in 3. 7 years, as measured by travelers on the spacecraft?

Answers

The constant velocity that the spacecraft must travel is : 3.49 * 10⁸ m/s

Given data :

Distance of star from earth = 4.3 light years

Observers time = 3.7 years

Determine the constant velocity the spacecraft must travel

Observers time = 3.7 * 365 * 24 * 60 * 60

Distance of star from earth = 4.3 * 9.46 * 10¹⁵

The velocity the spacecraft must travel will be calculated using the equation

V = distance / time

   = ( 4.3 * 9.46 * 10¹⁵ ) / ( 3.7 * 365 * 24 * 60 * 60 )

   = 3.49 * 10⁸ m/s

Hence we can conclude that The constant velocity that the spacecraft must travel is : 3.49 * 10⁸ m/s

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Attached below is the missing detail related to the question

The constant velocity must the spacecraft travels from the earth will be 3.49×10⁸ m/sec.

What is velocity?

The change of distance with respect to time is defined as speed. Speed is a scalar quantity. It is a time-based component. Its unit is m/sec.

The given data in the problem is

Distance of earth from stars = 4.3 light year

The given time = 3.7 year

One year converted into the second as;

[tex]\rm 1 \ year = 3.7 \times 365 \ days \ \times 24 \ hour \times 60 \ minute \times 60 \ sec \\\\ 1 year = 116 \times 10^6 \ sec[/tex]

1 light-year into a kilometer as;

[tex]\rm 4.3 \ light year = 4.3 \times 9.46 \times 10^{15 metre} \\\\[/tex]

At what constant velocity must a spacecraft travel from the earth if it is to reach the star in 3. 7 years,

The velocity of the spacecraft's travel from the earth will be;

[tex]\rm V = \frac{d}{t} \\\\ V= \frac{4.3 \times 9.46 \times 10^{15}}{3.7 \times 365 \times 24 \times 60 \times 60 } \\\\\ V= 3.49 \times 10^8 \ m/sec[/tex]

Hence the constant velocity must the spacecraft travels from the earth will be  3.49×10⁸ m/sec.

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A particle is moving in a circle of radius 2 meters according to the relation (please refer to picture), where Vo is measured in radians and in seconds. The speed of the particle at = 4 seconds is
(A) 13 m/s
(B) 16 m/s
(C) 26 m/s
(D) 52 m/s)
(E) 338 m/s

Answers

Answer:

ω = dΘ / dt

dΘ / dt = d (3 t^2 + 2 t) / dt = 6 t + 2

At 4 sec       ω = 6 * 4 + 2 = 26 / sec

V = ω * R = 26 / sec * 2 m = 52 m/s

A skateboarder initially has 5 kJ of kinetic energy. As she freewheels along a flat section of path, she does 400 J of work against friction and air resistance. Calculate her final kinetic energy.​

Answers

The final kinetic energy of the skateboarder after she freewheels and did work against friction on the flat section of the path is 4,600 J.

Conservation of energy

The final kinetic energy of the stakeboarder is determined by applying the principle of conservation of energy as shown below;

ΔK.E = -W

K.Ef - K.Ei = -W

where;

K.Ef is the final kinetic energyK.Ei is the initial kinetic energyW is work done

K.Ef = K.Ei - W

K.Ef = 5,000 J - 400 J

K.Ef = 4,600 J

Thus, the final kinetic energy of the skateboarder is 4,600 J.

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5)A 0.50 kg hockey puck is at rest on ice when you hit it with a hockey stick, applying a force of 100 N for
0.10 seconds. The puck then slides across the ice, where the coefficient of kinetic friction is 0.20. After
sliding for 4 seconds, the puck collides with a 0.80 kg box of donuts. The puck immediately comes to rest
after the collision. How far will the box of donuts slide before coming to rest if the coefficient of friction
between the box and the ice is 0.30.

Answers

Answer:

F t = m Δv         impulse delivered = change in momentum

Δv = 100 * .1 / .5 = 20 m/s     original speed of puck

KE = 1/2 m v^2 = .5 * 20^2 / 2 = 100 J     initial KE of puck

E = μ m g d        energy lost by puck

Ff = μ m g = m a      deceleration of puck due to friction

a = μ  g = 9.8 * .2 = 1.96 m/s^2

v2 = a t + v1 = -1.96 * 4 + 20 = 12.2 m/s     speed of puck on striking box

m v2 = M V       conservation of momentum when puck strikes box

V = m v2 / M = 12.2 * .5 / .8 = 7.63 m/s     speed of box after collision

KE = 1/2 M V^2 = .8 * 7.63^2 / 2 = 23.3 J     KE of box after collision

KE = μ M g d     energy lost by box in sliding distance d

d = 23.3 / (.3 * .8 * 9.8) = 9.91 m     distance box slides

The water was used to power a generator, creating _______________ energy

Answers

Answer:

electric

please give brainliest

Using the model above, which letter represents the position of the earth when the northern hemisphere is experiencing the shortest hours of daylight
A

B

C

D

Answers

Answer:

At position "B" the northern hemisphere is tilted the most from the earth-sun direction and will experience the shortest hours of daylight, (and also the most indirect sun rays).

The presence of which magnetic feature best explains why a magnet can act at a distance on other magnets or on objects containing certain metals? fields dipoles domains forces

Answers

The orientation with the largest magnetic torque on the dipole is Orientation B (See attached image).

What is Magnetic Torque?

The torque on the dipole is defined as:

τ = µ×B,

where B is the external magnetic field.

The magnitude of this torque is µB sinθ, where θ is the angle between B and µ

Magnetic Torque is highest when;

→     →

µ ⊥ β

That is when θ  = 90°. Hence B is the correct answer. Please see attached image.

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Answer:

b

Explanation:

Which Tesla would win in a drag race?

(A) 2009 Tesla Roadster

(B) 2016 Tesla Model X

Answers

B 2016 tesla model x

Answer:B

Explanation:  

A has about 248 hp  but B has about 683 hp. 2016 is younger too. good day

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