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
DaniilM [7]
2 years ago
8

Find a power series for the function, centered at c, and determine the interval of convergence. f(x) = 9 3x + 2 , c = 6

Mathematics
1 answer:
san4es73 [151]2 years ago
6 0

Answer:

\frac{9}{3x + 2} = 1 - \frac{1}{3}(x - \frac{7}{3}) + \frac{1}{9}(x - \frac{7}{3})^2 - \frac{1}{27}(x - \frac{7}{3})^3 ........

The interval of convergence is:(-\frac{2}{3},\frac{16}{3})

Step-by-step explanation:

Given

f(x)= \frac{9}{3x+ 2}

c = 6

The geometric series centered at c is of the form:

\frac{a}{1 - (r - c)} = \sum\limits^{\infty}_{n=0}a(r - c)^n, |r - c| < 1.

Where:

a \to first term

r - c \to common ratio

We have to write

f(x)= \frac{9}{3x+ 2}

In the following form:

\frac{a}{1 - r}

So, we have:

f(x)= \frac{9}{3x+ 2}

Rewrite as:

f(x) = \frac{9}{3x - 18 + 18 +2}

f(x) = \frac{9}{3x - 18 + 20}

Factorize

f(x) = \frac{1}{\frac{1}{9}(3x + 2)}

Open bracket

f(x) = \frac{1}{\frac{1}{3}x + \frac{2}{9}}

Rewrite as:

f(x) = \frac{1}{1- 1 + \frac{1}{3}x + \frac{2}{9}}

Collect like terms

f(x) = \frac{1}{1 + \frac{1}{3}x + \frac{2}{9}- 1}

Take LCM

f(x) = \frac{1}{1 + \frac{1}{3}x + \frac{2-9}{9}}

f(x) = \frac{1}{1 + \frac{1}{3}x - \frac{7}{9}}

So, we have:

f(x) = \frac{1}{1 -(- \frac{1}{3}x + \frac{7}{9})}

By comparison with: \frac{a}{1 - r}

a = 1

r = -\frac{1}{3}x + \frac{7}{9}

r = -\frac{1}{3}(x - \frac{7}{3})

At c = 6, we have:

r = -\frac{1}{3}(x - \frac{7}{3}+6-6)

Take LCM

r = -\frac{1}{3}(x + \frac{-7+18}{3}+6-6)

r = -\frac{1}{3}(x + \frac{11}{3}+6-6)

So, the power series becomes:

\frac{9}{3x + 2} =  \sum\limits^{\infty}_{n=0}ar^n

Substitute 1 for a

\frac{9}{3x + 2} =  \sum\limits^{\infty}_{n=0}1*r^n

\frac{9}{3x + 2} =  \sum\limits^{\infty}_{n=0}r^n

Substitute the expression for r

\frac{9}{3x + 2} =  \sum\limits^{\infty}_{n=0}(-\frac{1}{3}(x - \frac{7}{3}))^n

Expand

\frac{9}{3x + 2} =  \sum\limits^{\infty}_{n=0}[(-\frac{1}{3})^n* (x - \frac{7}{3})^n]

Further expand:

\frac{9}{3x + 2} = 1 - \frac{1}{3}(x - \frac{7}{3}) + \frac{1}{9}(x - \frac{7}{3})^2 - \frac{1}{27}(x - \frac{7}{3})^3 ................

The power series converges when:

\frac{1}{3}|x - \frac{7}{3}| < 1

Multiply both sides by 3

|x - \frac{7}{3}|

Expand the absolute inequality

-3 < x - \frac{7}{3}

Solve for x

\frac{7}{3}  -3 < x

Take LCM

\frac{7-9}{3} < x

-\frac{2}{3} < x

The interval of convergence is:(-\frac{2}{3},\frac{16}{3})

You might be interested in
A student is assessing the correlation between the number of hours a plant receives sunlight and the height to which it grows. T
IgorC [24]

Answer:

yes there is a linear correlation

reason : a increase of 1 hour results in an increase of 3 height

y= 3x + 4

it has  a positive slope of 3 ,and y intercept of 4

Step-by-step explanation:

7 0
3 years ago
Read 2 more answers
The regular price of a box of granola is 6.36. Today there is a special deal for 3.56 off each box. How much does a box of grano
stiks02 [169]

Answer:

with the special deal each box of granola would cost2.80

Step-by-step explanation:

6 0
2 years ago
2(2x − 5 − 4) = 160 − 17
wariber [46]
4x - 9 = 143

4x = 152

x = 38
3 0
3 years ago
Read 2 more answers
What is the positive solution of x2 – 36 = 5x?
goldenfox [79]

Answer:

9

Step-by-step explanation:

x² − 36 = 5x

x² − 5x − 36 = 0

(x − 9) (x + 4) = 0

x = 9 or -4

3 0
2 years ago
What percent of 50 is 10
gayaneshka [121]
50% of 10 is just half so half of 10 is 5 
The answer is five 
5 0
3 years ago
Read 2 more answers
Other questions:
  • {9-[8-(1/3+1/4)×6]}÷[8(1/3+1/2)×6]
    15·1 answer
  • This graph shows a proportional relationship between the number of tulips and the number of daffodils in gardens planned by the
    5·2 answers
  • PLEASE HELP I've tried solving this so many times I don't understand!
    6·1 answer
  • A trail mix has 4 cups of raisins, 3 cups of dates, 6 cups of peanuts, and 2 cups of cashews. Which ingredients are in the same
    5·1 answer
  • I need help solving this
    7·1 answer
  • A triangle has sides with lengths of 4 meters, 5 meters and 3 meters . Is it a right triangle? ​
    13·1 answer
  • I just need the numbers can anyone help me with this ??
    8·2 answers
  • Please answer correctly !!!!! Will mark Brianliest !!!!!!!!!!!!!!
    5·2 answers
  • One hundred draws are made at random with replacement from box A, and 250 are made at random with replacement from box B. (a) 50
    9·1 answer
  • 1. Which of the following is NOT a monomial?<br> 12xy<br> 2x + 9<br> -8x<br> 15
    5·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!