1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
arlik [135]
3 years ago
6

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

Mathematics
1 answer:
Inga [223]3 years ago
7 0

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

You might be interested in
Please Help Marking Brainiest
Volgvan

Answer:

(4,18)

Step-by-step explanation:

4 0
2 years ago
Read 2 more answers
Write an expression for the total cost of admission and r rides if an amusement park charges $5 for admission and $2 for each ri
vaieri [72.5K]
2r + 5

Hope this helps! :)
7 0
3 years ago
Read 2 more answers
Which of the following numbers has a prime factorization of 2 x 5 x 5?
Archy [21]

Answer:

5×5

Step-by-step explanation:

the factorazation is 5×5

7 0
3 years ago
Simplify 7 – (–12) – 5
kherson [118]

Answer:

14

Step-by-step explanation:

1. Do the "Two Mark" Rule

    -  7 + (+12) - 5

2. Do your addition before subtracting

    - 7 + 12 = 19

3. Subtract your remaining two numbers

   -  19 - 5 = 14

5 0
3 years ago
Use appropriate algebra and theorem 7.2.1 to find the given inverse laplace transform. (write your answer as a function of t.) ℒ
castortr0y [4]

We assume you want to find the inverse transform of s/(s^2 +3s -4). This can be written in partial fraction form as

(4/5)/(s+4) + (1/5)/(s-1)

which can be found in a table of transforms to be the transform of

(4/5)e^(-4t) + (1/5)e^t


_____

There are a number of ways to determine the partial fractions. They all start with factoring the denominator.

s^2 +3x -4 = (s+4)(s-1)


After that, you can postulate the final form and determine the values of the coefficients that make it so. For example:

A/(s+4) + B/(s-1) = ((A+B)s + (4B-A))/(s^2 +3x -4)


This gives rise to two equations:

(A+B) = 1

(4B-A) = 0

4 0
3 years ago
Other questions:
  • Przeciętny orzeł waży 1,68 kg, a przeciętny wróbel - 25 g. Mózg orła waży 14 g, a mózg wróbla - 1 g. Oblicz, jakim procentem mas
    9·1 answer
  • Problem 7-12 don't understand it so can I have answer please.
    5·1 answer
  • Use the slope of the given line to construct a parallel line
    13·2 answers
  • When contor lines are spaced evenly the slope of the hillside is?
    12·1 answer
  • B.? What number is 1/4 of 100?​
    6·1 answer
  • Round 47,250 to nearest thousand
    10·2 answers
  • Watch help video<br> Determine the x-intercepts of the following equation.<br> (1 – 2)(–2 – 5) = y
    12·1 answer
  • HELP ASAP!!!! Need to get grade up
    7·2 answers
  • Need help on this please
    11·1 answer
  • D(1) = 1,d(n) = n.d(n-1) for n ≥ 2
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!