Question

Find a power series for f(x)= (9x-3)/x²-1 centered at 0 by adding two power series.

Answer 1

To find the power series for f(x) = (9x-3)/(x^2-1) centered at 0, add the power series of 9x - 3 and the power series of 1/(x^2-1).

Step-by-step explanation:

To find the power series for f(x) = (9x-3)/(x^2-1) centered at 0, we will add two power series:

1. P(x) = 9x - 3 (power series with constant term -3)
2. Q(x) = 1/(x^2-1) (power series representation of the rational function)

First, let's find the power series representation for P(x) = 9x - 3:

• The constant term of the power series is -3.
• The coefficient of x^n in the power series is 9 for all n >= 1.

Next, let's find the power series representation for Q(x) = 1/(x^2-1):

• Using partial fraction decomposition, we can write Q(x) = 1/((x+1)(x-1)).
• Now we can use the geometric series expansion to represent Q(x) as a power series.

Finally, we can add the power series for P(x) and Q(x) to get the power series for f(x):

• f(x) = P(x) + Q(x) = (9x - 3) + Q(x)

This is the power series representation of f(x) centered at 0.