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
GenaCL600 [577]
2 years ago
7

Find a power series representation for the function. (give your power series representation centered at x = 0. ) f(x) = ln(5 − x

)
Mathematics
1 answer:
klio [65]2 years ago
3 0

Recall that for |x|, we have the convergent geometric series

\displaystyle \sum_{n=0}^\infty x^n = \frac1{1-x}

Now, for \left|\frac x5\right| < 1, we have

\dfrac1{5 - x} = \dfrac15 \cdot \dfrac1{1 - \frac x5} = \dfrac15 \displaystyle \sum_{n=0}^\infty \left(\frac x5\right)^n = \sum_{n=0}^\infty \frac{x^n}{5^{n+1}}

Integrating both sides gives

\displaystyle \int \frac{dx}{5-x} = C + \int \sum_{n=0}^\infty \frac{x^n}{5^{n+1}} \, dx

\displaystyle -\ln(5-x) = C + \sum_{n=0}^\infty \frac{x^{n+1}}{5^{n+1}(n+1)}

If we let x=0, the sum on the right side drops out and we're left with C=-\ln(5).

It follows that

\displaystyle \ln(5-x) = \ln(5) - \sum_{n=0}^\infty \frac{x^{n+1}}{5^{n+1}(n+1)}

or

\displaystyle \ln(5-x) = \boxed{\ln(5) - \sum_{n=1}^\infty \frac{x^n}{5^n n}}

You might be interested in
Factor the quadratic equation 0=15x squared + 39 x-18
leva [86]

Answer:

The factors of given quadratic equation

<em>   </em>x = \frac{9}{15} and x = -2<em />

Step-by-step explanation:

<u><em>Explanation:-</em></u>

<em>Given quadratic equation 15x² +39 x -18 =0</em>

<em> The factors of 270 = 30 × 9</em>

<em>The quadratic equation 15x² +39 x -18 =0</em>

<em>                                    ⇒ 15 x² +30 x-9x -18=0</em>

<em>                                 ⇒ 15x(x+2) -9( x+2) =0</em>

<em>                                ⇒( 15x -9)(x+2) = 0</em>

<em>                       15x - 9=0    and  x+2 =0</em>

<em>                       15 x =9   and  x = -2</em>

<em>                       </em>x = \frac{9}{15} and x = -2<em />

<u><em></em></u>

3 0
3 years ago
Which expression is equivalent to 15n – 20? HURRY PLEASE!!!
Nikolay [14]
The answer is C. 5(3n-4)
8 0
4 years ago
Read 2 more answers
Find the volume of the cylinder.<br> Either enter an exact answer in terms of it or use 3.14 for T.
kaheart [24]

Answer:

188 units cubed

Step-by-step explanation:

6 0
3 years ago
Read 2 more answers
Help me please top brainilest
Tatiana [17]

i think it would be your first choice

5 0
4 years ago
Read 2 more answers
There are 350 Families lives in the small town of America. A poll of 50 families revealed the mean annual church contribution is
timofeeve [1]

Answer:

$550+/-$20.79

= ( $529.21, $570.79)

Therefore, the 95% confidence interval (a,b) = ( $529.21, $570.79)

Step-by-step explanation:

Confidence interval can be defined as a range of values so defined that there is a specified probability that the value of a parameter lies within it.

The confidence interval of a statistical data can be written as.

x+/-zr/√n

Given that;

Mean x = $550

Standard deviation r = $75

Number of samples n = 50

Confidence interval = 95%

z(at 95% confidence) = 1.96

Substituting the values we have;

$550+/-1.96($75/√50)

$550+/-1.96($10.60660171779)

$550+/-$20.7889393668

$550+/-$20.79

= ( $529.21, $570.79)

Therefore, the 95% confidence interval (a,b) = ( $529.21, $570.79)

5 0
4 years ago
Other questions:
  • In the coordinate plane, quadrilateral ABCD has vertices with coordinates A(1,−1), B(−5,3), C(−3,6), and D(3,2). ​ ​Part A: Comp
    8·1 answer
  • Which numbers below belong to the inequality check all that apply x/6 greater than equal to 9
    9·1 answer
  • Simplify y 2 + 11y - 6y + y 2.<br><br> y4 + 5<br> y4 + 5y<br> 2y 2 + 5<br> 2y 2 + 5y
    5·1 answer
  • Can a base angle of an isosceles triangle be an obtuse angle
    10·1 answer
  • A system of equations is shown below. Which of the following statements describes the graph of this system of equations in the (
    5·1 answer
  • A landscaper is constructing a rectangular garden with an area of 108 square feet. He draws the garden on paper and represents t
    10·1 answer
  • Problem 2 Find all the divisors of the number 2,310. How do you know that you found them all?​
    9·1 answer
  • Amanda is hiking with her daughter, Mia who is only 1 yaar old and has very short leg. For every 3 steps Amanda makes, Mia takes
    14·1 answer
  • 4(1/4+3/4)+(-2)= what ​
    13·1 answer
  • Nikki spends
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!