Question

Show that the coefficient of

x^{n}

in the expansion of

\frac{x}{(x - a) (x - b)}

in ascending powers of

x

is

\frac{a^{n} - b^{n}}{a - b} \cdot \frac{1}{a^{n} b^{n}}

and that the coefficient of

x^{2 n}

in the expansion of

\frac{{(1 + x^{2})}^{n}}{{(1 - x)}^{3}}

is

2^{n - 1} (n^{2} + 4 n + 2)

.

Accepted Answer

As we know that ${(1 - x)}^{- n} = 1 + (- n) (- x) + \frac{(- n) (- n - 1)}{2!} {(- x)}^{2} + \dots$ , where $n \in ℝ$ and $|x| < 1$ .

We have $\frac{x}{(x - a) (x - b)} = \frac{1}{a - b} [\frac{a}{x - a} - \frac{b}{x - b}]$

$= \frac{1}{(a - b)} [- a {(a - x)}^{- 1} + b {(b - x)}^{- 1}]$

$= \frac{1}{a - b} [- {(1 - \frac{x}{a})}^{- 1} + {(1 - \frac{x}{b})}^{- 1}]$

$= \frac{1}{a - b} [(1 + \frac{x}{b} + \frac{x^{2}}{b^{2}} + \dots \frac{x^{n}}{b^{n}} + \dots) - (1 + \frac{x}{a} + \frac{x^{2}}{a^{2}} + \dots \frac{x^{n}}{a^{n}} + \dots)]$

$∴$ Coefficient of $x^{n} = \frac{1}{a - b} [\frac{1}{b^{n}} - \frac{1}{a^{n}}]$

$= \frac{1}{a - b} (\frac{a^{n} - b^{n}}{a^{n} \cdot b^{n}})$

Given $\frac{{(1 + x^{2})}^{n}}{{(1 - x)}^{3}} = \frac{{\{{(1 - x)}^{2} + 2 x\}}^{n}}{{(1 - x)}^{3}}$

$= \frac{1}{{(1 - x)}^{3}} [{(1 - x)}^{2 n} + n \cdot {(1 - x)}^{2 n - 2} 2 x + \dots + {}^{n}C_{n - 2} {(1 - x)}^{4} {(2 x)}^{n - 2} + {}^{n}C_{n - 1} {(1 - x)}^{2} {(2 x)}^{n - 1} + {(2 x)}^{n}]$

Thus $x^{2 n}$ comes from the last two terms i.e., $n (\frac{1}{1 - x}) {(2 x)}^{n - 1} + \frac{{(2 x)}^{n}}{{(1 - x)}^{3}} = n {(2 x)}^{n - 1} (1 + x + x^{2} + \dots) + {(2 x)}^{n} (1 + 3 x + \dots + \frac{1 \cdot 2 \cdot 3 \cdot \dots n (n + 1) (n + 2)}{2 \cdot n!} + \dots)$

$∴$ Coefficient of $x^{2 n} = n \cdot 2^{n - 1} + 2^{n} \frac{(n + 1) (n + 2)}{2}$

$= 2^{n - 1} [n + n^{2} + 3 n + 2]$

$= 2^{n - 1} [n^{2} + 4 n + 2]$

Hence, proved.

H S Hall and S R Knight Solutions for Chapter: Miscellaneous Examples, Exercise 1: Exercise 1

H S Hall Mathematics Solutions for Exercise - H S Hall and S R Knight Solutions for Chapter: Miscellaneous Examples, Exercise 1: Exercise 1

Questions from H S Hall and S R Knight Solutions for Chapter: Miscellaneous Examples, Exercise 1: Exercise 1 with Hints & Solutions

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