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
Two is equal to 4 less than 3 times a certain number. Find the number.
Allushta [10]

Answer:

2

Step-by-step explanation:

2=3x-4

6=3x

x=2

5 0
3 years ago
Read 2 more answers
What is the equation of the line that is parallel to the line y - 1 = 4(x + 3) and passes through the point (4, 32)?
adoni [48]

Answer:

4x- 16

Step-by-step explanation:

Heineken v River vs ievrroe wfej ghe

3 0
3 years ago
For a daily airline flight between two cities, the number of pieces of checked luggage has a mean of 380 and a standard deviatio
Olegator [25]
<span>Since one standard deviation is 20 luggages, 3 standard deviations above the mean is 3*20=60 luggages above the mean of 380 luggages, so 60+380 gives the answer C, 440.</span>
3 0
3 years ago
Read 2 more answers
1/4(x-2)+4=12 help please
ollegr [7]

Answer:

X = 34

Step-by-step explanation:

4 0
3 years ago
Please help thank you !
makkiz [27]
If he has 6 friends they will all get 8 cards
5 0
3 years ago
Read 2 more answers
Other questions:
  • Please help I will give you more points and brainlest!!!
    9·2 answers
  • If I am 3 miles below sea level what would the zero mean
    11·1 answer
  • #37 I need help with. best answer plz???
    5·1 answer
  • In AIJK, j = 940 inches, LJ=20° and ZK=50°. Find the length of i, to the nearest<br> inch.
    11·1 answer
  • To lower his insurance rates, Damon increases his deductible from $240 to $360. What is the precent increase in his deductible?
    14·1 answer
  • The distance from Leoti, Kansas to Dighton, Kansas on the map is approximately 2.5 inches. The key to the map states that 1 inch
    13·1 answer
  • King Soopers has two different sizes of sport drinks. Sigi wanted to make sure that he got the size with the better price. There
    11·1 answer
  • Answer please i need help
    5·2 answers
  • The distance across a two lane road
    6·1 answer
  • In which number does the 7 have a value that is 1/100 the value of the 7 in 468.705?
    10·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!