Express 1/(1 − x)2 as a power series by differentiating the equation below. what is the radius of convergence? 1 1 − x = 1 + x + x2 + x3 + = [infinity] xn n = 0 |x| < 1

S₂=1/(1-x)² = (1/x)∑ n*xⁿ

the power series converges also for |r|<1

Step-by-step explanation:

we know that the infinite geometric series

S= ∑ rⁿ  , from n=0 to n=∞

S= ∑ rⁿ = 1/(1-r)

then differentiating S with respect to r

dS/dr=d[1/(1-r)] = 1/(1-r)²

and

dS/dr=d(∑ rⁿ)/dr =∑ d(rⁿ)/dr  = ∑ n*rⁿ⁻¹ = (1/r)∑ n*rⁿ

then

S₂= (1/r)∑ n*rⁿ =  1/(1-r)²

thus

r²*S₂- r*S₂ = ∑ n*rⁿ⁺¹-∑ n*rⁿ = ∑ (n+1-1)*rⁿ⁺¹ - ∑ n*rⁿ  = (∑ (n+1)*rⁿ⁺¹ - ∑ n*rⁿ )- ∑rⁿ⁺¹ = (n+1)*rⁿ⁺¹ - (1-rⁿ⁺²)/(1-r)

r²*S₂- r*S₂ = r*S₂*(r-1) = [(n+1)*rⁿ⁺¹] - (1-rⁿ⁺²)/(1-r)

S₂ = (1-rⁿ⁺²)/[r*(1-r)²] - [(n+1)*rⁿ⁺¹ ]/[r*(1-r)]

thus |r|<1  in the first term to converge and for the second term for n→∞  , S₂= 1/(1-r)² → (n+1)*rⁿ⁺¹=n*rⁿ= 1 → ln r= -(ln n) /n

then knowing that lim (ln x)= ln (lim x)

lim (ln r)=  -lim [(ln n) /n ] = - lim ( 1/n) = 0 = ln r → r=1

therefore for the second term |r|< 1

then S₂ converges for |r|< 1

2/3

step-by-step explanation:

the slope of a line is equal to the rise/run. in this case the line rises 2 units for every +3 units it moves

6,250

step-by-step explanation:

to get the area, multiply 360 by 160.(57,600)

to find out how many bags of fertilizer it will take, first figure out how many square feet one bag can hold. divide 1000 by 8(125) to get the unit rate(how many square feet one pound of fertilizer can hold)and multiply it by 50(6250). if i haven't made a mistake, that is the answer.