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Find a generating function for the sequence of squares: g(x) = P∞ n=0 n 2x n . Then for fun, as above evaluate your expression g(1/100) or g(1/1000) to get a fraction that contains the squares in its decimal expansion.

Answers

Answer 1

Answer:

The sequence of squares, $\{n^2\}_(n\ge0)$ , has generating function

$g(x)=\displaystyle\sum_(n=0)^\infty n^2x^n$

Recall that for $|x|<1$ ,

$f(x)=\frac1{1-x}=\displaystyle\sum_(n=0)^\infty x^n$

Taking the derivative, we have

$f'(x)=\frac1{(1-x)^2}=\displaystyle\sum_(n=0)^\infty nx^(n-1)=\frac1x\sum_(n=0)^\infty nx^n$

and taking the derivative again, we have

$f''(x)=\frac2{(1-x)^3}=\displaystyle\frac1x\sum_(n=0)^\infty n^2x^(n-1)-\frac1{x^2}\sum_(n=0)^\infty nx^n$

$f''(x)=\frac2{(1-x)^3}=\displaystyle\frac1{x^2}\left(\sum_(n=0)^\infty n^2x^n-\sum_(n=0)^\infty nx^n\right)$

From this we can get an expression for $g(x)$ in terms of the derivatives of $f(x)$ :

$f''(x)=(g(x)-xf'(x))/(x^2)$

$\implies g(x)=x^2f''(x)+xf'(x)$

$\implies g(x)=(2x^2)/((1-x)^3)+\frac x{(1-x)^2}$

$\implies\boxed{g(x)=(x^2+x)/((1-x)^3)}$

Then

$g\left(\frac1{100}\right)=(10,100)/(970,299)\approx0.\underline{01}\,\underline{04}\,\underline{09}\,\underline{16}\ldots$

$g\left(\frac1{1000}\right)=(1,001,000)/(997,002,999)\approx0.\underline{001}\,\underline{004}\,\underline{009}\,\underline{016}\ldots$

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There are 5 strawberry boxes that are each filled with strawberries. Someone eats 5 of the strawberries. If each box holds a total of 18 strawberries, how many strawberries are left?

Answers

Well, 18-5=13, so there are 13 strawberries left.

There are 13 strawberries left.

What is the slope intercept equation of the line below. Y-intercept=(0,2) slope=3/4

Answers

Answer:

y = 3/4x +2

Step-by-step explanation:

The slope intercept form of an equation is

y = mx+b

where m is the slope and b is the y intercept

Y-intercept=(0,2) slope=3/4

y = 3/4x +2

What is 62 divided by 17 plus 29

I will mark brainliest

Answers

62/17=3.647059+29= 32.64059 but if you round just say 33!

Answer:

32.647

Step-by-step explanation:

1) Find the sum of 3/5 and 2/52) Find the difference between 2/3 and 1/4
3). what is 5 3/10 minus 2/5
4). For the school's inframurals a group of student prepared 23 1/12 liters of lemonade to sell at the end of the day they had 3 5/8 liters left over. how many liters of lemonade were sold?
5). my mother brought 3 3/4 kg of beef
2 3/5 kg of pork and 5 1/2 kg of chicken how many kilo grams of meat did she buy

Answers

Step-by-step explanation:

1) 3/5 + 2/5 = 5/5 or 1

2) to do this problem you need to make the denominators the same

(2/3)4 (1/4)3 8/12- 3/12. = 5/12

3)likewise here, you only have to change one denomimator

5 3/10 - (2/5)2

5 3/10 - 4/10

now to make this easiee for myself I prefer to make it a mixed number so 5(10)+ 3 = 53/10

53/10 - 4/10 = 49/10 then we convert this back to a mixed number and thus

5 3/10 - 2/5 = 4 9/10

4) 23 (1/12)2 - x = (3 5/8)3

23 2/24 - x = 3 15/24

- 3 15/24 +x = -3 15/24 + x

19 21/24 = x

simplify 19 7/8 liters sold

5) 3 3/4 + 2 3/5 + 5 1/2 = 11 17/20

Can someone help me and can y’all show me how you got the answer

Answers

Answer:

x = -8

Step-by-step explanation:

Step 1: Write equation

1/2x + 13 = 9

Step 2: Solve for x

Subtract 13 on both sides: 1/2x = -4
Multiply both sides by 2: x = -8

Step 3: Check

Plug in x to verify it's a solution.

1/2(-8) + 13 = 9

-4 + 13 = 9

9 = 9

Answer:

-8

Step-by-step explanation:

you use inverse operation

meaning opposite signs

subtract -13 from 13 cross it out

subtract 13 from 9

you get 1/2x=-4

divide 1/2 on both sides

-4 divided by 1/2 =-8

1.
WRITE "six million five hundred thousand four" in standard form.

Answers

for six million five hundred thousand four 6,500,004

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