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
amid [387]
3 years ago
5

Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interv

al. If the answer is an interval, enter your answer using interval notation. If the answer is a finite set of values, enter your answers as a comma separated list of values.)
Mathematics
1 answer:
sleet_krkn [62]3 years ago
4 0

Answer:

(0, 16]

Step-by-step explanation:

∑ₙ₌₁°° (-1)ⁿ⁺¹ (x−8)ⁿ / (n 8ⁿ)

According to the ratio test, if we define L such that:

L = lim(n→∞) |aₙ₊₁ / aₙ|

then the series will converge if L < 1.

aₙ = (-1)ⁿ⁺¹ (x−8)ⁿ / (n 8ⁿ)

aₙ₊₁ = (-1)ⁿ⁺² (x−8)ⁿ⁺¹ / ((n+1) 8ⁿ⁺¹)

Plugging into the ratio test:

L = lim(n→∞) | (-1)ⁿ⁺² (x−8)ⁿ⁺¹ / ((n+1) 8ⁿ⁺¹) × n 8ⁿ / ((-1)ⁿ⁺¹ (x−8)ⁿ) |

L = lim(n→∞) | -n (x−8) / (8 (n+1)) |

L = (|x−8| / 8) lim(n→∞) | n / (n+1) |

L = |x−8| / 8

For the series to converge:

L < 1

|x−8| / 8 < 1

|x−8| < 8

-8 < x−8 < 8

0 < x < 16

Now we check the endpoints.  If x = 0:

∑ₙ₌₁°° (-1)ⁿ⁺¹ (0−8)ⁿ / (n 8ⁿ)

∑ₙ₌₁°° -(-1)ⁿ (-8)ⁿ / (n 8ⁿ)

∑ₙ₌₁°° -(8)ⁿ / (n 8ⁿ)

∑ₙ₌₁°° -1 / n

This is a harmonic series, and diverges.

If x = 16:

∑ₙ₌₁°° (-1)ⁿ⁺¹ (16−8)ⁿ / (n 8ⁿ)

∑ₙ₌₁°° (-1)ⁿ⁺¹ (8)ⁿ / (n 8ⁿ)

∑ₙ₌₁°° (-1)ⁿ⁺¹ / n

This is an alternating series, and converges.

Therefore, the interval of convergence is:

0 < x ≤ 16

Or, in interval notation, (0, 16].

You might be interested in
Define $g$ by $g(x)=5x-4$. If $g(x)=f^{-1}(x)-3$ and $f^{-1}(x)$ is the inverse of the function $f(x)=ax+b$, find $5a+5b$.
Slav-nsk [51]

Answer:

<h2>2</h2>

Step-by-step explanation:

Given g(x)=5x-4 and g(x)=f^{-1}(x)-3

Substituting g(x) =  5x-4 into the second equation we have;

5x-4 =  f^{-1}(x)-3

f^{-1}(x) = 5x-4+3

f^{-1}(x) = 5x-1

To get f(x), let us first make y to be equal  f^{-1}(x)

y = 5x-1

expressing x in terms of y to get f(x), we have;

5x = y+1

x = y/5+1/5

replacing y with x, we will have;

y = x/5 + 1/5

F(x) = x/5 + 1/5

Comparing x/5 + 1/5 with ax+b, a = 1/5 and b = 1/5

5a + 5b = 5(1/5)+ 5(1/5)

5a+5b = 1+1

5a+5b = 2

8 0
3 years ago
The area of rectangular kitchen tile is 6x^2-8x-30. The width is 2x-6.
mrs_skeptik [129]

Answer:

L=6x^2-4x+5

P=12x^2-4x-2

Step-by-step explanation:

A=WL

So, we know the area is 6x^2-8x-30 and the width is 2x-6

Substitute

(6x^2-8x-30)=(2x-6)(L)

L=(6x^2-8x-30)/(2x-6

Divide

L=6x^2-4x+5

P=2L+2W

Substitute

P=2(6x^2-4x+5)+2(2x-6)

P=12×^2-8x+10+4x-12

Simplify

P=12x^2-4x-2

7 0
2 years ago
Twelve large bottles of water cost $9.How many bottles can you buy for $3?
Andrej [43]

Answer: 4 large bottles of water for $3.

Step-by-step explanation: To find how many large water bottles you can buy with $3, you’ll have to divide 12 by 3 which equals to 4. You can check your answer by multiplying 4 x 3 which equals to 12, which is the original amount of large water bottles you can buy for $9.

5 0
3 years ago
*****urgent help ****Write 2447457.627 in standard form to a appropriate degree of accuracy pls i need help
erica [24]

Answer:

2.447457627 x 10^6

Step-by-step explanation:

If this helped can i have Brainliest

5 0
3 years ago
Toss coin 3 times. What is probability of getting 1 head and 2 tails in any order?
Ugo [173]
1/8 because

1/2 x 1/2 x 1/2
7 0
3 years ago
Read 2 more answers
Other questions:
  • Use an appropriate substitution to solve the equation
    8·1 answer
  • What is the length?<br> The length of the pendulum is feet.<br> (Whole question above)⬆️
    15·1 answer
  • Any one wanna help with this question
    7·2 answers
  • Does the parabola with equation x2 + 5x +9=0 have real or imaginary roots?
    11·1 answer
  • Solve for x: 4 over x equals 5 over 10 8 9 20 40
    15·1 answer
  • Solve for x. 11x+4&lt; 15 OR 12x–7&gt;-25
    13·1 answer
  • Please help, I mark brainliest
    15·1 answer
  • A backpack regularly Price for $120 was marked down to $84. What was the backpack’s percent of discount?
    12·2 answers
  • I really need help on this:3 2/4+ 2 3/4+ 2 3/4+ 2 1/4+ 1 1/4+ 1
    7·1 answer
  • Solve the inequalities<br> (3x+5)(4x-8)&gt;0<br> (5-x)(2x+7)(x-3)≤0
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!