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
Artist 52 [7]
4 years ago
8

Find a compact form for generating functions of the sequence 1, 8,27,... , k^3

Mathematics
1 answer:
pantera1 [17]4 years ago
5 0

This sequence has generating function

F(x)=\displaystyle\sum_{k\ge0}k^3x^k

(if we include k=0 for a moment)

Recall that for |x|, we have

\displaystyle\frac1{1-x}=\sum_{k\ge0}x^k

Take the derivative to get

\displaystyle\frac1{(1-x)^2}=\sum_{k\ge0}kx^{k-1}=\frac1x\sum_{k\ge0}kx^k

\implies\dfrac x{(1-x)^2}=\displaystyle\sum_{k\ge0}kx^k

Take the derivative again:

\displaystyle\frac{(1-x)^2+2x(1-x)}{(1-x)^4}=\sum_{k\ge0}k^2x^{k-1}=\frac1x\sum_{k\ge0}k^2x^k

\implies\displaystyle\frac{x+x^2}{(1-x)^3}=\sum_{k\ge0}k^2x^k

Take the derivative one more time:

\displaystyle\frac{(1+2x)(1-x)^3+3(x+x^2)(1-x)^2}{(1-x)^6}=\sum_{k\ge0}k^3x^{k-1}=\frac1x\sum_{k\ge0}k^3x^k

\implies\displaystyle\frac{x+4x^3+x^3}{(1-x)^4}=\sum_{k\ge0}k^3x^k

so we have

\boxed{F(x)=\dfrac{x+4x^3+x^3}{(1-x)^4}}

You might be interested in
Find the value of x in each triangle. Please help me <33
frozen [14]

Answer:

x = 11

Step-by-step explanation:

The sum of the 3 angles in a triangle = 180°

sum the 3 angles and equate to 180

5x + 4 + 6x - 3 + 4x + 14 = 180 ← collect like terms on left side

15x + 15 = 180 ( subtract 15 from both sides )

15x = 165 ( divide both sides by 15 )

x = 11

6 0
3 years ago
Please help I need the answer right
Alenkinab [10]

Answer:

B) 4 inches

Step-by-step explanation:

Since ABC is congruent to EFG then AB is congruent to EF, and BC is congruent to FG.

BC = 4 in

FG = 4 in

5 0
3 years ago
Which ordered pair is a solution of the equation?<br> Y-4=7(x-6)
deff fn [24]

Answer:

(6,4)

Step-by-step explanation:

The equation is in 'point-slope' form.

y-y_1=m(x-x_1)\\\rule{150}{0.5}\\(x_1,y_1) \rightarrow \text{a point that's given}\\\\m \rightarrow \text{slope}

This means we can identify the point that was used in the equation.

y-\boxed{y_1} =m(x-\boxed{x_1}) | y-\boxed{4}=7(x-\boxed{6})\\\rule{210}{0.5}\\y_1 = 4\\\\x_1 =6\\\rule{210}{0.5}\\(x_1,y_1) \rightarrow \boxed{(6,4)}

(6,4) would be a solution to the equation given.

Hope this helps.

4 0
2 years ago
The linear equation c = 0.1998s + 76.4520 models the number of calories, c, in a beef hot dog as a function of the amount of sod
Karolina [17]

Answer:

The slope of the linear equation is 0.1998, and it means that the calories in a beef hot dog is increasing with the rate of 0.1998 calories per milligram of sodium.

Step-by-step explanation:

  • Using y = mx + b, we know that m represents the slope.
  • Looking at the equation of c = 0.1998s + 76.4520, our m is equal to 0.1998.
  • This slope means that the rate is increasing by 0.1998 calories for every milligram of sodium.
6 0
3 years ago
The temperature one afternoon at 3 PM
prohojiy [21]

Answer:

Your answer is 49 degrees

Step-by-step explanation:

66 - 17 = 49

7 0
3 years ago
Other questions:
  • 8. Find the radius of a circular lid with an area of 12in.^2.
    11·1 answer
  • Mia hired a moving company. The company charged
    10·1 answer
  • 832x275 to the greatest place value
    10·1 answer
  • Customer account numbers for a certain company consists of 3 letters followed by 5 numbers how many different account numbers ar
    8·1 answer
  • Rewrite this expression <br><img src="https://tex.z-dn.net/?f=%20%5Csqrt%5B5%5D%7Bx%7D%20" id="TexFormula1" title=" \sqrt[5]{x}
    15·1 answer
  • Find the surface area of the triangular prism below. The bases each have an
    9·1 answer
  • complete the ordered pain so that each is a solution of the given linear equation then graph the equation y=x+6​
    6·1 answer
  • HELP ASAP!! - Natalie is working two summer jobs, making $7 per hour babysitting and making $11 per hour clearing tables. In a g
    15·1 answer
  • The average reading score on certain tests is given by y = 0.153x + 255.4, where x is the number of years past 1970. In what yea
    11·1 answer
  • The length of the base of an isosceles triangle is x. The length of a leg is 4x - 7. The perimeter of the triangle is 31. Find x
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!