Question

The power series representation of the function f(x)=x/(4−x) is *​

Answer 1

Answer:

Step-by-step explanation:

We can calculate the derivatives directly. An alternate approach is to note that 4−x=4(1−x4). Thus

(4−x)−3=143(1−x4)−3.

Now we will find the power series expansion of (1−t)−3. Note that the second derivative of (1−t)−1 is 2(1−t)−3.

We know that the power series expansion of (1−t)−1 is 1+t+t2+t3+t4+⋯. So we differentiate twice, and put the pieces together.

Answer 2

Given,

f(x)=x/(4−x)

To find,

The power series representation of the function

Solution,

We can calculate the derivatives directly. An alternate approach is to note that 4−x=4(1−x4).

Thus,

(4−x)−3=143(1−x4)−3.

Now we will find the power series expansion of (1−t)−3. Note that the second derivative of (1−t)−1 is 2(1−t)−3.

We know that the power series expansion of (1−t)−1 is 1+t+t2+t3+t4+⋯. So we differentiate twice, and put the pieces together

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