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## Saturday, 9 August 2014

### CHAPTER 10 - Worked Out Examples

 Example: 11
 (a) For $|x|\; < \;1$ expand ${(1 - x)^{ - 3}}$ (b) Find the coefficient of ${x^n}$ in the expansion of${(1 + 3x + 6{x^2} + 10{x^3} +\ldots \infty )^{ - n}}$
 Solution: 11-(a)
We have
 ${V_r} = \dfrac{{( - 3)\;(( - 3) - 1)(( - 3) - 2)) \ldots (( - 3) - r + 1)}}{{r!}}$ $= \dfrac{{{{( - 1)}^r}\;(r + 2)!}}{{2(r!)}}$ $= \dfrac{{{{( - 1)}^r}\;(r + 1)\;(r + 2)}}{2}$
Thus,
 ${V_0} = 1,\;\;{V_1} = - 3,\;{V_2} = 6,\;{V_3} = - 10\ldots$
so that
 ${(1 - x)^{ - 3}} = 1 + 3x + 6{x^2} + 10{x^3} + \ldots \infty$
 Solution: 11-(b)
We use the result of part – $(a)$ in this:
 ${(1 + 3x + 6{x^2} + 10{x^3}\ldots \infty )^{ - n}} = {\left( {{{(1 - x)}^{ - 3}}} \right)^{ - n}}$ $= {(1 - x)^{3n}}$
The coefficient of ${x^n}$ in this binomial expansion (note: the power is now a positive integer) would be ${( - 1)^n} \cdot {\;^{3n}}{C_n}$.
 Example: 12
 Find the magnitude of the greatest term in the expansion of ${\left( {1 - 5y} \right)^{ - 2/7}}$for $y = \dfrac{1}{8}$.
 Solution: 12
Let us first do the general case: what is the greatest term in the expansion of ${(1 + x)^n}$, where $n$ is an arbitrary rational number. We have,
 ${T_{r + 1}} = {V_r}\;{x^r}$ and ${T_r} = {V_{r - 1}}\;{x^{r - 1}}$ so that $\dfrac{{{T_{r + 1}}}}{{{T_r}}} = \dfrac{{{V_r}}}{{{V_{r - 1}}}} \cdot x$ $= \dfrac{{n - r + 1}}{r} \cdot x$
Now, let us find the conditions for which this ratio exceeds $1$. We have
 $\left| {\;{T_{r + 1}}\;} \right|\; \ge \;\left| {\;{T_r}\;} \right|$ $\Rightarrow \left| {\;\dfrac{{n + 1}}{r} - 1} \right|\; \ge \;\dfrac{1}{{|x|}}$ $\ldots(1)$
For this particular problem, $(1)$ becomes
 $\left| {\;\dfrac{{\dfrac{{ - 2}}{7} + 1}}{r} - 1} \right|\; \ge \;\dfrac{1}{{\left| {\dfrac{{ - 5}}{8}} \right|}}$ $\Rightarrow\,\,\,\, \left| {\;\dfrac{5}{{7r}} - 1\;} \right|\; \ge \;\dfrac{8}{5}$ $\Rightarrow \,\,\,\,\dfrac{5}{{7r}}\; \ge \;\dfrac{{13}}{5}$ $\Rightarrow\,\,\,\, r \le \;\dfrac{{25}}{{91}}$ $\Rightarrow\,\,\,\, r=0$
Thus, ${T_{r + 1}} = {T_1}$ is the greatest term, with magnitude $1$