Please Express f(x)=(x+2)/(2x^2 - x - 1) as the sum of a power series by using partial fractions.

Also find interval of convergence.I would really appreciate the help.

Since f(x) = (x + 2)/((2x + 1)(x - 1)), partial fractions yields
(x + 2)/((2x + 1)(x - 1)) = A/(2x + 1) + B/(x - 1).

To solve for A and B, clear the denominators:
x + 2 = A(x - 1) + B(2x + 1).

Letting x = 1 yields 3 = 0 + 3B ==> B = 1.
Letting x = -1/2 yields 3/2 = -3A/2 + 0 ==> A = -1.
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So, f(x) = -1/(2x + 1) + 1/(x - 1).

Rewriting this in terms of geometric series:
f(x) = -1/(1 - (-2x)) - 1/(1 - x)
......= -Σ(n = 0 to ∞) (-2x)^n - Σ(n = 0 to ∞) x^n
......= Σ(n = 0 to ∞) [(-2)^(n+1) - 1] x^n.

Since the individual geometric series converge for |-2x| < 1 and |x| < 1,
we conclude that f converges on their intersection, which is |x| < 1/2.

I hope this helps!