Determine if the sequence {an} converges, and if it does, find its limit when an = (1 − 1/6n) ^5n

Answers

Answer 1

The sequence {an} converges to 1.

To determine if the sequence {an} converges, we can use the limit definition of convergence. Taking the limit as n approaches infinity of an, we have:

lim(n→∞) an = lim(n→∞) (1 − 1/6n) ^5n

Using the limit law for exponents, we can rewrite this as:

lim(n→∞) (1 − 1/6n) ^5n = [lim(n→∞) (1 − 1/6n)]^5n

Now we can use the limit law for products to separate the limit into two parts:

lim(n→∞) (1 − 1/6n) ^5n = [lim(n→∞) (1 − 1/6n)]^ [lim(n→∞) 5n]

The limit of (1 − 1/6n) as n approaches infinity is 1, so the first part simplifies to:

lim(n→∞) (1 − 1/6n) ^5n = 1^ [lim(n→∞) 5n]

The limit of 5n as n approaches infinity is infinity, so the second part is:

lim(n→∞) (1 − 1/6n) ^5n = 1^∞

This is an indeterminate form, so we need to use another method to find the limit. Taking the logarithm of both sides, we have:

ln(lim(n→∞) (1 − 1/6n) ^5n) = ln(1^∞)

Using the limit law for logarithms, we can rewrite this as:

lim(n→∞) 5n ln(1 − 1/6n) = ln(1)

The limit of ln(1 − 1/6n) as n approaches infinity is 0, so the left-hand side simplifies to:

lim(n→∞) 5n ln(1 − 1/6n) = 0

This means that the limit of the sequence {an} is 1, since:

lim(n→∞) an = lim(n→∞) (1 − 1/6n) ^5n = 1^∞ = e^0 = 1

Therefore, the sequence {an} converges to 1.

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Related Questions

please help and be correct please :( have a explanation also

Answers

Answer:

Part A: 3 bowls    Part B: See Below

Step-by-step explanation:

Part A: If he uses 1/6 to make ONE bowl, 3/6 is one half so he can make THREE bowls.

Part B: Just draw the fact of 1/2 being equal to 3/6 there for making 3 bowls and you'll be good! :)

Hope this Helped! ^ - ^

You conduct a simulation regarding students scores on a physics test. Your null hypothesis is
that the mean score on the test is less than or equal to 80 points. Your alternative hypothesis is
that the mean score is greater than 80 points.
You determine that the results of your simulation have a p-value of 0.06. What does this mean?

Answers

Under the tested hypothesis, the outcome of the simulation has a probability of 6%. With a significance level of 5%, you should FAIL to reject the null hypothesis.

The p-value is the probability of getting true null hypothesis.

What is Null Hypothesis and Alternative Hypothesis?

Null hypothesis is a hypothesis of a test which assumes that there is no statistical relationship between two variables. This is the hypothesis which is being testes.

Alternative hypothesis assumes that there is a relationship between two variables.

Given that,

You conduct a simulation regarding students scores on a physics test.

Null hypothesis is that the mean score on the test is less than or equal to 80 points.

Alternative hypothesis is that the mean score is greater than 80 points.

p value = 0.06 = 6%

This means that there is only a 6% that the null hypothesis of yours is true.

So the p value tells you the probability of getting a true null hypothesis.

Hence the p value is the probability of getting true null hypothesis.

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how does (46)(4) = 56 please tell me

Answers

Answer:

ok, so 46 times 4 is 184.

Step-by-step explanation:

46

x 4

-------

184

So 6 times 4 is 24 and then you carry over the 2.  4 times 4 is 16.  You add 2 to 16 and you get 18.  So the total would be 184.

It's kinda hard to explain over brainly

Answer:

impossible

Step-by-step explanation:

46x4= 276

please mark me as brainlist

please hurry! I'm so confused

Answers

Answer:

Step-by-step explanation:

The general equation is

y = 2.49x + 24.99

The number of cones bought is x

The cake cost is 24.99

The total cost is y

===========

So the meaning of (5,37.44) is that she bought 5 cones and 1 cake. Let's just check that out.

Check

y = 2.49*5 + 24.99

y = 12.45 + 24.99

y = 37.44

So the total cost (y) of 5 cones (2.49 each) and 1 cake ($24.99) is 37.44

(help) Evaluate the expression 9y - 4(4) + y3 when y = 2.

Answers

Answer:

9y - 4(4) + y³= 18-16+8=10

Can someone please help me!!! Ill mark brainliest!!

Answers

Answer: 252 ft

Find the area of the whole thing, then find the area of the green rectangle, then subtract to find the answer.

Pepper has 7.5 bags to clear 12 yd squared to distribute evenly among the beds . About how many bags will she need to use for each square yard?

Answers

Answer:

5/8 bags

Step-by-step explanation:

Pepper ha 7.5 bags

These bags are needed to clear 12 yards squared equally

Therefore the number of bags required for each square yard can be calculated as follows

= 7.5/12

= 0.625

0.625 to fraction is 5/8

Hence 5/8 bags is needed for each square yard

Solve for y. 1 y – 1 3y + 7 10 y E​

Answers

Answer:

[tex]y = \frac{17}{7}[/tex]

Step-by-step explanation:

1) To solve this question easily, cross multiply first. Multiply the denominator of the first fraction by the numerator of the second fraction, and multiply the denominator of the second fraction by the numerator of the first fraction:

[tex]\frac{y-1}{3y+7}=\frac{1}{10}\\1(3y+7) = 10(y-1)\\3y + 7 = 10y-10[/tex]

2) Now, solve like you would and isolate y. Subtract both sides by 10y.

[tex]3y + 7 = 10y-10\\-7y + 7 = -10[/tex]

3) Then, subtract both sides by 7.

[tex]-7y + 7 = -10\\-7y = -17\\[/tex]

4) Finally, divide both sides by -7 to receive the answer.

[tex]-7y = -17\\y = \frac{-17}{-7} \\y = \frac{17}{7}[/tex]

please help me on this will give you brainliest

Answers

Answer:

The answer is really simple. Add 33 and 67 then subtract it from 180 and the answer is 80.

Answer:

x=80 degrees

Step-by-step explanation:

upper right corner angle =67 degrees because it is a corresponding congruent angle.

The sum of all the inside angles of all triangles is 180 degrees.

180-33-67=x

x=80 degrees

2x x 5z x y
simplify the equation ​

Answers

Answer: The simplified form of 2x · 5z · y is 10xzy.

Step-by-step explanation:

Garrett needs 3 3 gallons of fruit juice to make punch. He already has 56 5 6 gallon of grape juice and 23 2 3 gallon of orange juice. Drag amounts of juice that he can add to the juice he already has to make exactly 3 3 gallons. Juice Garrett has. Grape juice is labeled five-sixths gallon. Orange juice is labeled two-thirds gallon. CLEAR CHECK Orange juice labeled one-third gallon. Green juice is labeled one-sixth gallon. Juice Needed

Answers

Answer:

no por que ella no te ama lo siento

what do you observe about the form of the equation of any circle whose center is at the point of origin​

Answers

Ce clasă ești sa văd daca te pot ajuta

Drew has more than 4 times as many baseball cards as mark. Katie has fewer than half as many cards as mark. Mark has m baseball cards

Answers

Answer:

Cards with Mark = 30

Cards with Drew  = 120

Cards with Katie = 12

Step-by-step explanation:

The complete question is

Drew has 4 times as many baseball cards as Mark. Katie has 3 fewer than than half as many cards  as Mark. In all they have 162 baseball cards. How many does each have?

Solution -

Let the number of cards with Mark be "X"

Number of cards with Drew = 4 *X = 4X

Number of Cards with Katies = 0.5 X -3

Sum of all the cards is equal to 162

X+4X + 0.5 X -3 = 162

5.5 X = 165

X = 30

Cards with Mark = 30

Cards with Drew = 4 * 30 = 120

Cards with Katei = 15-3 = 12

Geometry help, please!!!
provide the missing reasons for the proof.
Given: Prove: ΔABC ~ ΔDBE
_____________________________________________
picture provided below

Answers

Answer:

1: Given

2: Verical angles are always congruent

3: Angle Angle theorem

HELPPP. i need help with this i have trouble with ordered pairs

Answers

the second one i got (0,-2) first one i got (0,9)

what is slope of (12.-18) and (-15.-18).​

Answers

Answer:

3

Step-by-step explanation:

The slope (m) is 0.
The formula you’ll use is y2-y1/x2-x1

A snail is moving along a path that is 4 meters long. The snail moves at an average rate of 3 3/10 inches per minute. Question 1 Part A Find the length of the path to the nearest inch. (Hint: 1 in.≈2.54 cm) Enter the correct answer in the box.

Answers

Answer: The length of the path = 157 inches { approx}

Step-by-step explanation:

Given: The length of the path = 4 meters

1 meter = 100 centimeters

So 4 meters = 400 centimeters

Also 1 in. ≈ 2.54 cm

[tex]\Rightarrow\ 1 cm = \dfrac{1}{2.54}\ in.[/tex]

Now, [tex]\text{ 400 centimeters}=400\times\dfrac{1}{2.54}\text{ inches}[/tex]

[tex]=157.480314961\approx 157 \ \ \ \ \ \text{ [Rounded to the nearest inch.]}[/tex]

Hence, the length of the path = 157 inches { approx}

Find the value of 200 + 9 + 0.3 +0.001.​

Answers

Answer:

209.301

Step-by-step explanation:

45 is 3% of what number?

quick I need an answer!
hurry pls!

Answers

Answer:

stop rushing brat

Step-by-step explanation:

Step-by-step explanation:

Steps to solve "45 is 3 percent of what number?"

We have, 3% × x = 45

or,

3

100

× x = 45

Multiplying both sides by 100 and dividing both sides by 3,

we have x = 45 ×

100

3

x = 1500

If you are using a calculator, simply enter 45×100÷3, which will give you the answer.

Question Progress
Solve the inequality 5y + 4 < 3y + 17

Answers

Answer:

y < [tex]\frac{13}{2}[/tex]

Step-by-step explanation:

Given

5y + 4 < 3y + 17 ( subtract 3y from both sides )

2y + 4 < 17 ( subtract 4 from both sides )

2y < 13 ( divide both sides by 2 )

y < [tex]\frac{13}{2}[/tex]

Answer:

y < 13/2

Step-by-step explanation:

1/3 + 3/8 =?? wich method can sole the number sentence​

Answers

Answer: 17/24
Eplination: you need a common demonator

i really need help with this

Answers

Answer:

A+6+A = 20

A = 7

Step-by-step explanation:

J = number of problems Juana completed

A = number of problems Andy completed

J+A=20

J = A + 6

Replace J with A+6 in the first equation

A+6+A = 20

2A +6 = 20

Subtract 6 from each side

2A +6-6 = 20-6

2A =14

Divide by 2

2A/2 = 14/2

A = 7

Answer:

7

Step-by-step explanation:

set up your two equations.

j + a = 20. equation 1

j = a + 6. equation 2

since you know that j = a + 6, you can substitute a + 6 into the place of j. this is because you want to isolate a in order to find out how many problems andy completed.

j + a = 20

a + 6 + a = 20

now solve.

2a + 6 = 20

2a = 14

a = 7

andy has completed 7 problems. if juana completed 6 more than andy, then she has completed 13. 13 + 7 = 20.

What do these circles have in common?

Answers

They are all circles

Answer:

They are all circles? I dont really know how to explain it so that is all I got to say

Step-by-step explanation:

Enter 2/5 and 1/3 as a pair of equivalent fractions with the least common denominator between the two fractions.

Answers

Answer:

2/5 = 6/15

1/3 = 5/15

Step-by-step explanation:

LCD of 5 and 3 is 15.

So 2/5 = 2*3/15 = 6/15

and

1/3 = 5*1 / 15 = 5/15

PLEASE HELP
7x+3y=20 and -4x-6y=11 find the value of 3x-3y

Answers

Answer:

im pretty sure the answer is A 6

Step-by-step explanation:

What number goes in the blank? (-3)(?)=36 *

Answers

Answer:

-12

Step-by-step explanation:

-3x=36

x=36/-3

x=-12

Answer:

-12

Because (-3)(-12)=36

any negative number multiplied by another = positive number :3

In the system of equations, which variable would it be easiest to solve for?

2 x + 4 y = 26. 3 x + y = 9.
The easiest to solve for is x in the first equation.
The easiest to solve for is y in the first equation.
The easiest to solve for is x in the second equation.
The easiest to solve for is y in the second equation.

Answers

Answer:

The easiest way of solving this is to eliminate one of the variables.  If you multiply each of the equations by a number so you can add the equations together and have only one variable left, you can solve the single variable equation.  Try multiplying the first equation by 11, the second by -10 and adding them, thus eliminating the y  variable.  After this, substitute that value you have for x and solve for y.  I hope this helps.

Step-by-step explanation:

D Itś D DDDDDD DDDDDDDDDDDDDDDDDD

Han's family got a solar panel. Each month they get a credit to their account for the electricity that is generated by the solar panel. The credit they receive varies based on how sunny it is. In January they used $83.56 worth of electricity and generated $6.75 worth of electricity. Here is their electricity bill from January. Current charges: $83.56 Solar Credit: -$6.75 Amount due: $76.81 1) In July they were traveling away from home and only used $19.24 worth of electricity. Their solar panel generated $22.75 worth of electricity. What was their amount due in July?

Answers

Answer:

-$3.51

Step-by-step explanation:

Given that :

Current charges:

$83.56 Solar Credit: -$6.75

Amount due: $76.81

July they were traveling away from home and only used $19.24 worth of electricity. Their solar panel generated $22.75 worth of electricity. What was their amount due in July

Worth of electricity used in July = $19.24

Worth of electricity generated by solar = - $22.75

Electricity used in July + solar credit charge

$19.24 - $22.75

= -$3.51

(9r^2+4r-7)+(3r^2-3r)

Answers

Answer:

12r2+r−7

Step-by-step explanation:

hope it helped

Express two-thirds of age gap in terms of y if the age gap between a father and a daughter is y years

Answers

Answer:

(2/3)y

Step-by-step explanation:

Given that :

Age gap between father and daughter = y

2/3 of the difference in their ages expressed as in terms of y ;

2/ 3 * y

(2/3)y

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