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2 years ago

Will give brainleist

Mathematics

1 answer:

Natalka [10]2 years ago

8 0

Answer:

A.) 3n+m+2p

Step-by-step explanation:

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Use the Integral Test to determine whether the series is convergent or divergent

Inga [223]

Answer:

A. $\sum_{n=1}^{\infty}\frac{n}{e^{15n}}$ converges by integral test

Step-by-step explanation:

A. At first we need to verify that the function which the series is related ( $\frac{n}{e^{15n}}$ ) fills the necessary conditions to ensure that the test is effective.

*f(x) must be continuous or differentiable

*f(x) must be positive and decreasing

Let´s verify that $f(x)=\frac{n}{e^{15n}}$ fills these conditions:

*Considering that eˣ≠0 for all x, the function $f(x)=\frac{n}{e^{15n}}$ does not have any discontinuities, so it´s continuous

*Because eˣ is increasing:

if a<b ,then eᵃ<eᵇ

if 0<eᵃ<eᵇ ,then 1/eᵃ > 1/eᵇ

if 1/eᵃ > 1/eᵇ and a<b, then a/eᵃ<b/eᵇ

We conclude that $f(x)=\frac{n}{e^{15n}}$ is decreasing

*Because eˣ is always positive and the sum is going from 1 to ∞, this show that $f(x)=\frac{n}{e^{15n}}$ is positive in [1,∞).

Now we are able to use the integral test in $f(x)=\frac{n}{e^{15n}}$ as follows:

$\sum_{n=1}^{\infty}\frac{n}{e^{15n}}\ converges\ \leftrightarrow\ \int_{1}^{\infty}\frac{x}{e^{15x}}\ dx\ converges$

Let´s proceed to integrate f(x) using integration by parts

$\int_{1}^{\infty}\frac{x}{e^{15x}}\ dx=\int_{1}^{\infty}xe^{-15x}\ dx$

Choose your U and dV like this:

$U=x\ \rightarrow dU=1\\ dV=e^{-15x}\ \rightarrow V=\frac{-e^{-15x}}{15}$

And continue using the formula for integration by parts:

$\int_{1}^{\infty}Udv = UV|_{1}^{\infty} - \int_{1}^{\infty}Vdu$

$\int_{1}^{\infty}xe^{-15x}\ dx= \frac{-x}{15e^{15x}}|_{1}^{\infty} -\frac{-1}{15} \int_{1}^{\infty}e^{-15x}\ dx$

$\int_{1}^{\infty}xe^{-15x}\ dx= \frac{-x}{15e^{15x}}|_{1}^{\infty} -\frac{-1}{15}(\frac{-1}{15e^{15x}})|_{1}^{\infty}$

$\int_{1}^{\infty}xe^{-15x}\ dx= \frac{-x}{15e^{15x}}|_{1}^{\infty} -\frac{1}{225e^{15x}}|_{1}^{\infty}$

Because we are dealing with ∞, we´d rewrite it as a limit that will help us at the end of the integral:

$\int_{1}^{\infty}xe^{-15x}\ dx= \lim_{b \to{\infty}}(\frac{-x}{15e^{15x}}|_{1}^{b}-\frac{1}{225e^{15x}}|_{1}^{b})$

$\int_{1}^{\infty}xe^{-15x}\ dx= \lim_{b \to{\infty}} \frac{-b}{15e^{15b}}-\frac{1}{225e^{15b}}-(\frac{-1}{15e^{15}}-\frac{1}{225e^{15}})$

$\int_{1}^{\infty}xe^{-15x}\ dx= ( \lim_{b \to{\infty}} \frac{-b}{15e^{15b}}-\frac{1}{225e^{15b}})+\frac{1}{15e^{15}}(1-\frac{1}{15})$

We only have left to solve the limits, but because b goes to ∞ and it is in an exponential function on the denominator everything goes to 0

$\lim_{b \to{\infty}} \frac{-b}{15e^{15b}}-\frac{1}{225e^{15b}} = 0$

$\int_{1}^{\infty}xe^{-15x}\ dx= \frac{1}{15e^{15}}(1-\frac{1}{15})$

Showing that the integral converges, it´s the same as showing that the series converges.

By the integral test $\sum_{n=1}^{\infty}\frac{n}{e^{15n}}$ converges

7 0

2 years ago

3. Given this quadratic equation, 2x2 - x - 28 = 0,

JulijaS [17]

Answer:

a) (2x+7)(x-4)

b) x = -7/2 and x = 4

Step-by-step explanation:

a) (2x ?)(x ?)

set up the above and considered factors of 28 that, when paired with 2, would give me -1x and -28 → (1·28, 2·14, 4·7)

after trial-and-error, found that 4 and 7 worked (used un-FOIL)

b) 2x + 7 = 0

2x = -7

x = -7/2

*********

x-4 = 0

x = 4

4 0

2 years ago

Sadie wants to invest $2,000 in an account that earns 5.2% interest, compounded

hammer [34]

Answer:

29 years

Step-by-step explanation:

5.2% of 2000= 104 each year time 104 * 29 = 3016+2000=5016

2000 being the previous amount.

5 0

2 years ago

Three pounds of lawn seed covers 1800 square feet how many 4 pound bags are needed to cover 8400 square feet.

VikaD [51]

You need to find the unit rate of 3 : 1800 first by dividing 1800 and 3

1800 ÷ 3 = 600

So this means for every 1 pound bag it covers 600 square feet. You now need to multiply 600 and 4 to see how many square feet 4 pounds cover.

600 × 4 = 2400

So each 4 pounds covers 2400 sq ft. Now you have to divide 8400 and 2400 to find how many bags of 4 pounds you need

8400 ÷ 2400= 3.5

You need 3.5 bags of 4 pounds.
Technically you would need 4 bags because you’d have to round it up.

Hope this helps you!! :)

7 0

3 years ago

What is the importance of the rule of law<br>

V125BC [204]

Answer: The importance of the rule of law with these principles is self-evident. In our system, it is the foundation of governance. Under the rule of law, citizens can live and work safely. They will not be subjected to arbitrary government and are protected against any abuse of power by the state. hope this helps. Can u give me brainliest

Step-by-step explanation:

8 0

3 years ago

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