By using the power series for , find the power series for .

Options:

∑₂°° n (n − 1) xⁿ⁻²

Step-by-step explanation:

f(x) = 1 / (1 − x)

f'(x) = 1 / (1 − x)²

f''(x) = 2 / (1 − x)³

Therefore:

g(x) = f''(x)

g(x) = d²/dx² [1 / (1 − x)]

Using sum of a geometric series:

g(x) = d²/dx² ∑₀°° xⁿ

g(x) = d/dx ∑₁°° n xⁿ⁻¹

g(x) = ∑₂°° n (n − 1) xⁿ⁻²

use the power rule  

( a b ) n = a n b n

to distribute the exponent.

apply the product rule to  

3 x 2 y 3 . ( 3 x 2 ) 2 ( y 3 ) 2

apply the product rule to  

3 x 2 . 3 2 ( x 2 ) 2 ( y 3 ) 2

raise  

3

to the power of  

2 . 9 ( x 2 ) 2 ( y 3 ) 2 multiply the exponents in   ( x 2 ) 2

.

apply the power rule and multiply exponents,  

( a m ) n = a m n . 9 x 2 ⋅ 2 ( y 3 ) 2

multiply  

2   by   2 . 9 x 4 ( y 3 ) 2

multiply the exponents in  

( y 3 ) 2 .

apply the power rule and multiply exponents,  

( a m ) n = a m n . 9 x 4 y 3

⋅

2

multiply  

3   by   2 . 9 x 4 y 6

step-by-step explanation:

i need the options

step-by-step explanation: