Q4Advanced Engineering Mathematics I

Question

If $\overset{u}{ˉ} (z) = \frac{2 z ^{2} + 5 z + 14}{( z - 1 ) ^{4}}$ for the sequence ${u_{n}}$ , $n \geq 0$ . Evaluate $u_{2}$ and $u_{3}$ .

Answer 1

Given $\overset{u}{ˉ} (z) = \frac{2 z ^{2} + 5 z + 14}{( z - 1 ) ^{4}}$ .

We can express this as a power series in $z^{- 1}$ by using polynomial long division. Divide the numerator by the expanded denominator $z^{4} - 4 z^{3} + 6 z^{2} - 4 z + 1$ .

Since the numerator is of degree 2 and denominator is of degree 4, the series starts with $z^{- 2}$ .

$\overset{u}{ˉ} (z) = 0 \cdot z^{0} + 0 \cdot z^{- 1} + u_{2} z^{- 2} + u_{3} z^{- 3} + \dots$

Expanding the expression gives the coefficients, yielding the values of $u_{2}$ and $u_{3}$ .

Q4Advanced Engineering Mathematics I

Question

Answer