Question

If \( |x| < 1 \), the coefficient of \( x^2 \) in the power series expansion of \( \frac{x^4}{(x+1)(x-2)} \) is:

• A. 3
• B. 0
• C. -1
• D. -3

Hint:

For coefficient problems, expand step by step and isolate the required term.

Answer 1

The Correct Option is B

We are tasked with finding the coefficient of \(x^2\) in the power series expansion of: \[ f(x) = \frac{x^4}{(x + 1)(x - 2)} \] Step 1: Partial Fraction Decomposition We'll start by performing partial fraction decomposition on the given expression. \[ f(x) = \frac{x^4}{(x + 1)(x - 2)} \] By partial fraction decomposition, \[ f(x) = \frac{A}{x + 1} + \frac{B}{x - 2} \] Multiplying both sides by \((x + 1)(x - 2)\): \[ x^4 = A(x - 2) + B(x + 1) \] Expanding both sides: \[ x^4 = A(x - 2) + B(x + 1) \] \[ x^4 = A(x - 2) + B(x + 1) \] Expanding each term: \[ x^4 = A(x) - 2A + B(x) + B \] Equating coefficients, \[ x^4 = (A + B)x + (-2A + B) \] Step 2: Power Series Expansion We'll expand each term as a power series. Recall that: \[ \frac{1}{x + 1} = \sum_{n=0}^{\infty} (-1)^n x^n \] \[ \frac{1}{x - 2} = \sum_{n=0}^{\infty} 2^n x^{-n} \] Now combine these series expansions and identify the coefficient of \(x^2\). Step 3: Identify the Coefficient of \(x^2\) From the series expansions, the coefficient of \(x^2\) is 0. Step 4: Final Answer 

\[Correct Answer: (2) \ 0\]

Answer 2

To find the coefficient of \( x^2 \) in the expansion of \( \frac{x^4}{(x+1)(x-2)} \), we begin by simplifying and expanding the function.
First, decompose the function into partial fractions:
\[\frac{x^4}{(x+1)(x-2)}=\frac{A}{x+1}+\frac{B}{x-2}\]
Multiply throughout by the denominator \((x+1)(x-2)\) to clear the fractions:
\[x^4=A(x-2)+B(x+1)\]
Expand and simplify:
\[x^4=Ax-2A+Bx+B\]
\[x^4=(A+B)x+(-2A+B)\]
Since \(x^4\) has no other terms, compare and solve:
\[A+B=0\quad (1)\]
\[-2A+B=0\quad (2)\]
Add equations (1) and (2):
\[A+B-2A+B=0+0\]
\[-A+2B=0\]
From (1):
\[A=-B\]
Substitute \(A=-B\) into equation (2):
\[-2(-B)+B=0\]
\[2B+B=0\]
\[3B=0\]
\[B=0\] and hence \(A=0\)
This leads the function to \(\frac{x^4}{(x+1)(x-2)}\sim 0\)
Thus, it's evident that the coefficient of \(x^2\) is 0, as no terms contribute to \(x^2\) in the expansion.
Therefore, the coefficient of \(x^2\) is 0.

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