Question

Express the function f(x) =2x-4/x2-4x+3 as the sum of a power series by first using partial fractions. Find the interval of convergence.

Given that d / dx(1/ 1+3x)=- 3 /(1+3x)2' find a power series representation for 1 g(x)=-3/91+3x)2 by first representing f(x) = 1/1+3x as a power series, then differentiating term-by-term.

Answer 1

To express the function f(x) = 2x-4/(x^2-4x+3) as a power series, first factor the denominator and use partial fractions. The power series representation is f(x) = 1/(x-1) + 1/(x-3). The interval of convergence is (-∞, 1) ∪ (1, 3) ∪ (3, ∞).

Step-by-step explanation:

To express the function f(x) = 2x-4/(x^2-4x+3) as the sum of a power series, we first need to factor the denominator and use partial fractions:

x^2-4x+3 = (x-1)(x-3)

Then, we can write the function as:

f(x) = 2x-4/((x-1)(x-3)) = A/(x-1) + B/(x-3)

Now, we can find the values of A and B by equating the numerators:

2x-4 = A(x-3) + B(x-1)

Solving this system of equations, we get A = 1 and B = 1. Therefore, f(x) can be expressed as:

f(x) = 1/(x-1) + 1/(x-3)

To find the interval of convergence for the power series representation, we need to find the values of x for which the series converges. In this case, the interval of convergence is (-∞, 1) ∪ (1, 3) ∪ (3, ∞).

Answer 2

To express f(x) = 2x-4/x²-4x+3 as a power series using partial fractions, we first factor the denominator and apply partial fractions decomposition. Then, we express each fraction as a power series and combine the power series representations.

Step-by-step explanation:

To express the function f(x) = 2x-4/x²-4x+3 as a power series using partial fractions, we first factor the denominator as (x-1)(x-3). Then, we apply partial fractions decomposition: f(x) = A/(x-1) + B/(x-3), where A and B are constants. We can then find the values of A and B by equating the numerators of the fractions on the right-hand side with the numerator of the original function.

Once we have the partial fraction decomposition, we can express each fraction A/(x-1) and B/(x-3) as a power series using the formula 1/(1-x) = 1 + x + x²+ x³ + ... for |x| < 1. Finally, we combine the power series representations to obtain the power series representation of f(x).