(a) Use differentiation to find a power series representation for f(x) = 1 (9 + x)2 . f(x) = [infinity] n = 0 What is the radius of convergence, R? R = (b) Use part (a) to find a power series for f(x) = 1 (9 + x)3 . f(x) = [infinity] n = 0 What is the radius of convergence, R? R = (c) Use part (b) to find a power series for f(x) = x2 (9 + x)3 . f(x) = [infinity] n = 2 What is the radius of convergence, R? R =

Check the explanation

Step-by-step explanation:

Differentiation enables us or gives us the room to find rates of change. Now If y = some function of x (in another term if y is equal to an expression comprising of numbers and x's), then we can agree that the derivative of y (with respect to x) is written dy/dx, pronounced as "dee y by dee x" .

Kindly check the below image to get the step by step explanation to the question above.

(a)

(b)

(c)

All of them for

Step-by-step explanation:

To begin with remember what is the power series representation of

(a)

When we find the derivative of the function at the left side of the equality we get.

So basically our representation would be

Now if you reset the index by setting when  n=1   k=0, and if then . Therefore we will have

That converges for

Therefore is valid for

(b)

To begin with remember that

for

Therefore finding the derivative of the right expression and multiplying by -1/2 we get that

for

(c)

For this problem you just have to multiply to the result that you found in (b)

So you would get that

But once again to make it look more elegant you can adjust the index of the sum by setting , therefore if k=0 then n=2 and k=n-2 so we would have

once again for