write the equation in spherical coordinates. (a) x2 + y2 + z2 = 25

According to the question we have the solution of the  given differential equation initial value problem is: g(x) = (3/4)x^4 - x + 9/4 .

To solve the given initial value problem, we need to integrate both sides of the differential equation. We have:

g'(x) = 3x(x^2 - 1/3)

Integrating both sides with respect to x, we get:

g(x) = ∫[3x(x^2 - 1/3)] dx

g(x) = ∫[3x^3 - 1] dx

g(x) = (3/4)x^4 - x + C

where C is the constant of integration.

To find the value of C, we use the initial condition g(1) = 2. Substituting x = 1 and g(x) = 2 in the above equation, we get:

2 = (3/4)1^4 - 1 + C

2 = 3/4 - 1 + C

C = 9/4

Therefore, the solution of the given initial value problem is:

g(x) = (3/4)x^4 - x + 9/4

In more than 100 words, we can say that the given initial value problem is a first-order differential equation, which can be solved by integrating both sides of the equation. The resulting function is a family of solutions that contain a constant of integration. To find the specific solution that satisfies the initial condition, we use the given value of g(1) = 2 to determine the constant of integration. The resulting solution is unique and satisfies the given differential equation as well as the initial condition.

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Answer: 70π

Step-by-step explanation:

2xπx5x2 = 20π

2xπx25=50π

20π+50π=70π

Step-by-step explanation:

log3  x - 2125 = 3

log3  x = 2128

x = 3^(2128)      

   ( I think you need to check your post ! Format, syntax and parentheses are important!)

To find the other zeros of f(x), we can use polynomial division to divide f(x) by (x+1), since we know that -1 is a zero. This will give us a quadratic equation which we can solve using the quadratic formula. The polynomial division is:

     x^2 + 10x + 50
 ---------------------
x+1 | x^3 + 11x^2 + 60x + 50
    -x^3 - x^2
    ------------
          10x^2 + 60x
          -10x^2 - 10x
          ------------
                  50x
                  -50

So we have (x+1)(x^2 + 10x + 50) = 0. The quadratic equation x^2 + 10x + 50 = 0 has no real solutions, since its discriminant is negative. Therefore, the other two zeros of f(x) are complex conjugates of each other. We can use the quadratic formula to find them:
x = (-10 ± √(-300))/2 = -5 ± 5i√3
Thus, the zeros of f(x) are -1, -5 + 5i√3, and -5 - 5i√3.

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The correct statement about the electromagnetic spectrum is (d) the frequency of infrared radiation, which has a wavelength of 1.0 × 10³ nm, is twice that of radio waves, which have a wavelength of 1.0 × 10⁶ nm.

The speed of light in a vacuum is constant, so the speed of different types of electromagnetic waves is the same. Therefore, statement (a) is incorrect because the speed of x-rays and infrared radiation is the same.

The wavelength of visible radiation decreases as its color changes from red to violet, not from blue to green. Thus, statement (b) is incorrect.

The frequency of a wave is inversely proportional to its wavelength. Since infrared radiation has a shorter wavelength (1.0 × 10³ nm) compared to radio waves (1.0 × 10⁶ nm), it has a higher frequency. Therefore, statement (c) is incorrect.

On the other hand, statement (d) is correct because a shorter wavelength corresponds to a higher frequency. Thus, the frequency of infrared radiation (1.0 × 10³ nm) is indeed twice that of radio waves (1.0 × 10⁶ nm) due to the significant difference in their wavelengths.

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the distance from the base of the stand to the campfire is 285.6 feet.

The angle of depression of 35 degrees.

Let's denote the distance from the base of the stand to the campfire as "x."

Using the tangent function, we have:

tan(35 degrees) = opposite/adjacent

tan(35 degrees) = 200/x

To find the value of x, we can rearrange the equation:

x = 200 / tan(35 degrees)

x ≈ 200 / 0.7002

x ≈ 285.6 feet

Therefore, the distance from the base of the stand to the campfire is 285.6 feet.

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Express f(n) in Big O notation for the given recurrence relation by comparing log b(a) with log a/log b and applying the Master Theorem.

To find an expression for f(n) in Big O notation for the given recurrence relation f(n) = a f(n/b) c(n^(log a/log b)), where f(1) = 1, and a > 1 and b > 1, we can apply the Master Theorem.

The Master Theorem states that if a recurrence relation has the form f(n) = a f(n/b) + O(n^d), then the solution for f(n) can be expressed as:

If log b(a) > d, then f(n) = O(n^(log b(a))).

If log b(a) = d, then f(n) = O(n^d * log(n)).

If log b(a) < d, then f(n) = O(n^d).

In our case, we have f(n) = a f(n/b) c(n^(log a/log b)). By comparing the form of the given relation with the Master Theorem, we can see that log b(a) = log a/log b, and d = log a/log b.

Now, let's consider the three cases:

If log b(a) > d, then f(n) = O(n^(log b(a))).

If log b(a) = d, then f(n) = O(n^d * log(n)).

If log b(a) < d, then f(n) = O(n^d).

To determine which case applies, we need to compare log b(a) with log a/log b.

Finally, we can express f(n) in Big O notation based on the corresponding case determined above.

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The periodic transfer of a portion of the cost of an intangible asset to expense is known as amortization. This is the process of spreading the cost of an intangible asset over its useful life, similar to how depreciation is used for tangible assets like buildings and equipment.

Intangible assets, such as patents, copyrights, and trademarks, do not have a physical existence but still have value to the company. Amortization recognizes the decline in value of these assets over time and helps to accurately reflect their impact on the company's financial statements.

The amount of amortization each period is calculated by dividing the cost of the asset by its estimated useful life. It is important for companies to track and properly account for their intangible assets, including amortization, as it can have a significant impact on their financial statements and overall financial health.

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The complex power is 1800 VA.

What is the equivalent expression?

Equivalent expressions are expressions that perform the same function despite their appearance. If two algebraic expressions are equivalent, they have the same value when we use the same variable value.

To find the complex power (S) given the apparent power (q) and power factor (pf), we can use the following formulas:

S = q × pf

Given that q = 2000 var and pf = 0.9 (leading), we can substitute these values into the formula to calculate the complex power.

S = 2000 var × 0.9

S = 1800 VA (volt-ampere)

Therefore, the complex power is 1800 VA.

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No,  The given table does not show a proportional relationship.

We have to given that;

The table shown the value of x and y.

Since, We know that;

The proportion relation is,

y = kx

Where, k is constant of proportion.

By given table,

x = 0, y = 0

x = 16, y = 64

x = 63, y = 315

Hence, We get;

For x = 16, y = 64

k = 64/16

k = 4

For x = 63, y = 315

k = 315 / 63

k = 5

Hence, The given table does not show a proportional relationship.

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C - 1,4,4

D - 2,2,4

The product of dimensions must give 16 as a result.

Taking the square root of both sides, we get: y = ±sqrt(C)(x^2 + 78). This is the final solution to the differential equation. Note that we have included a constant of integration, which could take on any value and would affect the specific solution to the equation.

In this problem, we are given an equation that needs to be separated into variables, with x-terms on one side and y-terms on the other. We also need to avoid placing dy or dx in a denominator. Following this strategy, we can rewrite the original equation as:
y dy = x/(x^2 + 78) dx
Next, we need to integrate each side of the differential equation. Ignoring the constant of integration, we can integrate the left-hand side of the equation as follows:
∫ y dy = 1/2 y^2
To integrate the right-hand side of the equation, we can use the substitution u = x^2 + 78, which gives us du/dx = 2x and dx = du/2x. Substituting this back into the original equation, we get:
∫ x/(x^2 + 78) dx = ∫ 1/u du
The integral of 1/u is ln|u| + C, where C is the constant of integration. Substituting back for u, we get:
∫ x/(x^2 + 78) dx = ln|x^2 + 78|/2 + C
Putting this all together, we get:
1/2 y^2 = ln|x^2 + 78|/2 + C
Multiplying both sides by 2 and exponentiating, we get:
y^2 = Ce^(2ln|x^2 + 78|)
Simplifying this expression, we get:
y^2 = C(x^2 + 78)^2
Taking the square root of both sides, we get:
y = ±sqrt(C)(x^2 + 78)
This is the final solution to the differential equation. Note that we have included a constant of integration, which could take on any value and would affect the specific solution to the equation.

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The perpendicular distance from point P to the line connecting stations Shore and Rock is approximately 668,389.33 feet.

To find the perpendicular distance from point P to the line connecting stations Shore and Rock, we can use the formula for the distance between a point and a line.

The equation of the line connecting stations Shore and Rock can be determined using the slope-intercept form of a straight line: y = mx + b, where m is the slope and b is the y-intercept.

First, let's calculate the slope of the line:

slope = (Y2 - Y1) / (X2 - X1)

     = (392,133.86 - 394,087.52) / (652,531.72 - 654,127.26)

     = -1,953.66 / -1,595.54

     ≈ 1.224

Next, we can find the y-intercept (b) by substituting the coordinates of either station (e.g., Rock) into the slope-intercept form and solving for b:

392,133.86 = 1.224 * 652,531.72 + b

b ≈ 392,133.86 - 799,247.25

b ≈ -407,113.39

So, the equation of the line connecting Shore and Rock is:

y ≈ 1.224x - 407,113.39

Now, let's calculate the perpendicular distance from point P to the line using the formula:

distance = |Ax + By + C| / sqrt([tex]A^2[/tex] + [tex]B^2[/tex])

where A, B, and C are the coefficients of the line equation in the form Ax + By + C = 0. In this case, the equation of the line can be rewritten as:

-1.224x + y + 407,113.39 = 0

Therefore, A = -1.224, B = 1, and C = 407,113.39. Plugging in the coordinates of point P (653,594.81, 393,436.47) into the formula, we get:

distance = |-1.224 * 653,594.81 + 1 * 393,436.47 + 407,113.39| / sqrt((-1.224)^2 + 1^2)

        = |-799,103.63 + 393,436.47 + 407,113.39| / sqrt(1.497)

        = |1,001,446.23| / 1.225

        ≈ 818,003.79 / 1.225

        ≈ 668,389.33

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The component form of the resultant vector is ⟨3, 4⟩.

To find the component form of the resultant vector, we need to perform scalar multiplication and vector subtraction.

Let's start by performing the scalar multiplication:

1/3 = 1/3 × ⟨-9, 6⟩

= ⟨-9/3, 6/3⟩

= ⟨-3, 2⟩

Next, we perform vector subtraction:

-2 = -2 × (3, 1)

= ⟨-23, -21⟩

= ⟨-6, -2⟩

Now, we can find the resultant vector by subtracting the two vectors we obtained:

Resultant vector = 1/3- 2

= ⟨-3, 2⟩ - ⟨-6, -2⟩

= ⟨-3 - (-6), 2 - (-2)⟩

= ⟨-3 + 6, 2 + 2⟩

= ⟨3, 4⟩

Therefore, the component form of the resultant vector is ⟨3, 4⟩.

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The real zeros of the polynomial y = [tex]x^3 + 24x^2 + 144[/tex] using numerical method are approximately x ≈ -12.536; x ≈ -11.732 and x ≈ -0.732.

To find a possible formula for the polynomial with the given properties, we can start by considering the three roots: -2, 5, and 8. Since f(x) has a degree of three, we know that it can be written in the form:

f(x) = a(x - r1)(x - r2)(x - r3)

where r1, r2, and r3 are the roots, and 'a' is a constant.

Given that the roots are -2, 5, and 8, we have:

f(x) = a(x + 2)(x - 5)(x - 8)

Now, we need to find the value of 'a' to satisfy the condition f(7) = 9. Substituting x = 7 into the equation, we get:

9 = a(7 + 2)(7 - 5)(7 - 8)

9 = a(9)(2)(-1)

9 = -18a

Solving for 'a', we find:

a = -9/18

a = -1/2

Thus, a possible formula for the polynomial f(x) is:

f(x) = (-1/2)(x + 2)(x - 5)(x - 8)

Now, let's find the real zeros of the polynomial y = [tex]x^3 + 24x^2 + 144[/tex]

Setting y = 0, we have:

[tex]x^3 + 24x^2 + 144[/tex] = 0

To find the real zeros, we can use numerical methods or factoring. However, upon simplifying the equation, we can observe that it does not factor easily and does not have rational roots.

Using numerical methods, we can find the approximate real zeros:

x ≈ -12.536; x ≈ -11.732; x ≈ -0.732

Therefore, the real zeros of the polynomial y = [tex]x^3 + 24x^2 + 144[/tex] are approximately -12.536, -11.732, and -0.732.

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Given that $X$ has an exponential distribution with a mean of 10.

(a) We need to find $P(X<5)$. The cumulative distribution function (CDF) of the exponential distribution is given by $F(x) = 1 - e^{-\lambda x}$, where $\lambda$ is the rate parameter of the distribution. Since the mean of the distribution is 10, we have $\lambda = 1/10$. Therefore, we can write:

$P(X<5) = F(5) = 1 - e^{-\lambda \cdot 5} = 1 - e^{-1/2} \approx 0.3935$

(b) We need to find $P(15 < X < 2012)$. Again using the CDF of the exponential distribution, we can write:

= $P(15 < X < 2012) = F(2012) - F(15)

= (1 - e^{-\lambda \cdot 2012}) - (1 - e^{-\lambda \cdot 15})

= e^{-\lambda \cdot 15} - e^{-\lambda \cdot 2012} \approx 0.9997$

(c) From parts (a) and (b), we see that $P(X<5)$ is much smaller than $P(15 < X < 2012)$. This is because the exponential distribution has the memoryless property, which implies that the probability of an event occurring in the next $x$ units of time is independent of how much time has already elapsed. In other words, the distribution has no memory of past events.

Therefore, the probability of an event occurring in a short period of time is much smaller than the probability of it occurring in a longer period of time, even if the longer period of time starts after the short period.

In this case, the probability of $X$ being less than 5 is much smaller than the probability of $X$ being between 15 and 2012, even though 2012 is much larger than 15, because the exponential distribution "forgets" the past and treats each time interval independently.

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The value of the integral is:

∫[0 to 3] (1 + t^2) j + (4t^3)/(1 + t^4) k dt = 12 j + ln(82) k.

To evaluate the integral ∫[0 to 3] (1 + t^2) j + (4t^3)/(1 + t^4) k dt, we can compute the integral component-wise.

For the j-component:

∫[0 to 3] (1 + t^2) dt

Integrating term by term, we have:

∫[0 to 3] dt + ∫[0 to 3] t^2 dt

= [t] evaluated from 0 to 3 + [(1/3) t^3] evaluated from 0 to 3

= (3 - 0) + (1/3)(3^3 - 0^3)

= 3 + 9

= 12

For the k-component:

∫[0 to 3] (4t^3)/(1 + t^4) dt

Making a substitution u = 1 + t^4, du = 4t^3 dt, we have:

∫[1 to 82] (1/u) du

= ln|u| evaluated from 1 to 82

= ln|82| - ln|1|

= ln(82)

Therefore, the value of the integral is:

∫[0 to 3] (1 + t^2) j + (4t^3)/(1 + t^4) k dt = 12 j + ln(82) k.

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There are 6 club members who are both left-handed and like jazz music.

Let's use a Venn diagram to solve this problem. We have two sets: left-handed club members and club members who like jazz music.

We know that there are 20 club members in total. Out of these, 8 are left-handed and 15 like jazz music.

Now, let's fill in the information we have:

The left-handed circle will have 8 members.

The jazz music circle will have 15 members.

We also know that 2 club members are right-handed and dislike jazz music. Since people are either left-handed or right-handed, but not both, these two members cannot be in the left-handed circle. Therefore, they must be outside both circles.

Now, we can calculate the number of club members who are both left-handed and like jazz music by subtracting the number of club members in the right-handed and dislike jazz music category from the total number of left-handed club members:

Left-handed and like jazz music = Total left-handed - Right-handed and dislike jazz music

= 8 - 2

= 6

Therefore, there are 6 club members who are both left-handed and like jazz music.

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To find two positive numbers that satisfy the requirements of the given problem, we need to determine the values of the first and second numbers. The sum of the first number squared and the second number is 60, and we need to find the values that maximize their product.

Let's denote the first number as x and the second number as y. According to the problem, we have the equation x^2 + y = 60. To find the values that maximize the product xy, we can use optimization techniques. One approach is to use the AM-GM inequality, which states that the arithmetic mean of two positive numbers is always greater than or equal to their geometric mean.

Applying the AM-GM inequality, we have (x^2 + y)/2 ≥ √(x^2 * y). Simplifying this inequality, we get x^2 + y ≥ 2√(x^2 * y). Since the left side of the inequality is fixed at 60, the maximum value of the product xy occurs when equality is achieved in the AM-GM inequality.

Therefore, to find the values of x and y that maximize the product xy, we solve the equation x^2 + y = 60 and simultaneously satisfy the condition 2√(x^2 * y) = 60. By solving these equations, we can determine the values of the first and second numbers that satisfy the given requirements.

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We can write the matrix as:

A = [tex]\left[\begin{array}{ccc}-6&-5\\-6&-7\end{array}\right][/tex]

What is the eigenvalue?

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a nonzero vector that, when the linear transformation is applied to it, changes at most by a scalar factor.

Here, we have

Given the eigen values and eigen vectors of a 2×2 matrix A.

λ₁ = -4,    v₁ = [tex]\left[\begin{array}{r}-1&2\end{array}\right][/tex]

λ₂ = -1,    v₂ = [tex]\left[\begin{array}{r}-1&1\end{array}\right][/tex]

Here we have to find the matrix A.

So we have the system of equations:

Av₁ = λ₁ v₁

Let A = [tex]\left[\begin{array}{ccc}a&b\\c&d\end{array}\right][/tex]

Now, we have

[tex]\left[\begin{array}{ccc}a&b\\c&d\end{array}\right][/tex] [tex]\left[\begin{array}{r}-1&2\end{array}\right][/tex] = -4 [tex]\left[\begin{array}{r}-1&2\end{array}\right][/tex]

-a + 2b = -4....(1)

-c + 2d = -8...(2)

Similarly, we can write,

Av₂ =  λ₂v₂

[tex]\left[\begin{array}{ccc}a&b\\c&d\end{array}\right][/tex] [tex]\left[\begin{array}{r}-1&1\end{array}\right][/tex] = -1 [tex]\left[\begin{array}{r}-1&1\end{array}\right][/tex]

-a + b = 1...(3)

-c + d = -1...(4)

So, by solving equations (1) and (3), we get

(1) - (3)

b = -5

From(1), we get

-a + 2(-5) = -4

-a - 10 = -4

-a = 6

a = -6

Similarly, by solving equations (2) and (4) we get

(2) - (4) → d = -7

From(2), we get

-c + 2(-7) = -8

-c = -8 + 14

c = -6

Hence, we can write the matrix as:

A = [tex]\left[\begin{array}{ccc}-6&-5\\-6&-7\end{array}\right][/tex]

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the vector (3,4) can be expressed as (4)(-1,1) + (7/4)(4,0) = (-4,4) + (7,0) = (3,4).

The set S={(3,4), (-1,1), (4,0)} is linearly dependent.

To express the vector (3,4) as a linear combination of the vectors (-1,1) and (4,0), we need to find scalars (coefficients) x and y such that x(-1,1) + y(4,0) = (3,4).

Setting up the equations, we have:

-1x + 4y = 3

1x + 0y = 4

From the second equation, we can solve for x and get x = 4. Substituting this value into the first equation, we have:

-4 + 4y = 3

4y = 7

y = 7/4

Therefore, the vector (3,4) can be expressed as (4)(-1,1) + (7/4)(4,0) = (-4,4) + (7,0) = (3,4).

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You can say with 95% confidence that the true proportion of smokers in the population lies within the range of 12% plus or minus 0.16, or between 11.84% and 12.16%.

Suppose that you randomly selected 26 adults from the population. Assuming that 12% of the population smokes, you can calculate the expected number of smokers in your sample by multiplying the sample size by the population percentage:

26 x 0.12 = 3.12

Therefore, you would expect to find about 3.12 smokers in your sample. Since you cannot have a fraction of a person, you would round this answer to the nearest whole number, giving you an expected count of 3 smokers.

To determine the margin of error for this estimate, you can use the formula:

Margin of error = 1.96 x sqrt(p(1-p)/n)

where p is the population proportion (0.12), n is the sample size (26), and 1.96 is the z-score corresponding to a 95% confidence level.

Plugging in the values, you get:

Margin of error = 1.96 x sqrt(0.12 x 0.88/26) = 0.1586

Rounding this to two decimal places, the margin of error is 0.16.

Therefore, you can say with 95% confidence that the true proportion of smokers in the population lies within the range of 12% plus or minus 0.16, or between 11.84% and 12.16%.

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Answer: 211,900

Step-by-step explanation:

First you have to simplify both sides of the equation. Starting on the right, 2,654 x 100 is 265,400. On the right, 35,000 + 50,000 is 85,000.

Now you have 85,000 + x = 265,400. All you have to do is subtract 85,000 from both sides.

This gives you x = 211,900.

To find the first four terms of the given recursively defined sequence, we need to apply the given formula repeatedly. Starting with b1 = 1, Therefore, the first four terms of the sequence are: 1, 8, 96, 1536.

We Have: b2 = b1 * 4 * 2 = 8
b3 = b2 * 4 * 3 = 96
b4 = b3 * 4 * 4 = 1536
Therefore, the first four terms of the sequence are: 1, 8, 96, 1536.
We can observe that the sequence grows very quickly as k increases, since each term is multiplied by 4k. This is an example of an exponential growth, where the value of each term increases exponentially with the index k.
It's important to note that the given formula only works for integer values of k ≥ 2, since it involves raising 4 to the power of k. If k were a non-integer or negative value, the formula would not make sense and the sequence would not be well-defined.

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The number of text messages are the costs of the two different plans the same is 1250

We are given that;

() = { 19, ℎ ≤ 250

19 + 0.12( ― 250), ℎ > 250

Cost per cost=$45

Now,

For the second plan, the cost is $45 per month for unlimited text messages. That means each text message costs $45 / 30 = $1.50 / day. If you send 1250 text messages in a month, then each text message costs $1.50 / 1250 = $0.0012.

To find out when the costs are equal, we can set up an equation:

19 + 0.18(x - 250) = 45

where x is the number of text messages.

Solving for x gives:

x = 1250

Therefore, by the function the answer will be 1250.

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Answer:

45mm²

Step-by-step explanation:

area of parallelogram = base X height

= 5mm X 9mm

= 45(mm²)

To determine the p-value corresponding to a z-test statistic of 1.47 in a right-tailed test, we need to find the probability of obtaining a z-value equal to or greater than 1.47.

The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated under the null hypothesis.

Using a standard normal distribution table or calculator, we can find the area to the right of 1.47. This area represents the probability of obtaining a z-value greater than 1.47.

Looking up the z-score of 1.47 in a standard normal distribution table, we find that the corresponding area is approximately 0.9292.

Since this is a right-tailed test, the p-value is equal to the area to the right of the test statistic. Therefore, the p-value is approximately 0.9292.

Thus, the p-value is approximately 0.9292 or 92.92%.

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since the population standard deviations (σ1 and σ2) are not provided, we cannot calculate the exact test statistic Z.

To test the null hypothesis H0: (μ1 - μ2) = 0 versus the alternative hypothesis Ha: (μ1 - μ2) ≠ 0, we can use a two-sample z-test. The test statistic is calculated as:

Z = (x bar1 - x bar2) / sqrt((σ1^2 / n1) + (σ2^2 / n2))

Where:
X bar1 and x bar2 are the sample means of the two groups,
σ1 and σ2 are the population standard deviations of the two groups,
n1 and n2 are the sample sizes of the two groups.

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So the required blanks for the series are filled with:

Blank 1: 4ke⁻ᵏ

Blank 2: ((k + 1)/k)e⁻¹

Blank 3: <

Blank 4: The series is convergent.

The given series is,

[tex]\sum_{k=1}^{\infty}[/tex] 4ke⁻ᵏ

So the k th term of the series is given by,

aₖ = 4ke⁻ᵏ

Now,

aₖ₊₁/aₖ = (4(k+1)e⁻⁽ᵏ⁺¹⁾)/(4ke⁻ᵏ) = ((k + 1)/k)e⁻¹

Now the value of the limit is given by,

[tex]\lim_{k \to \infty}[/tex] |aₖ₊₁/aₖ| = [tex]\lim_{k \to \infty}[/tex] ((k + 1)/k)e⁻¹ = [tex]\lim_{k \to \infty}[/tex] (1 + 1/k)e⁻¹ = (1 + 0)e⁻¹ = e⁻¹

since e > 2

then e⁻¹ < 1/2

So, e⁻¹ < 1

So, it is less than 1.

since  [tex]\lim_{k \to \infty}[/tex] |aₖ₊₁/aₖ| = e⁻¹ < 1

Hence the given series [tex]\sum_{k=1}^{\infty}[/tex] 4ke⁻ᵏ is convergent.

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