Examples On Binomial Theorem For Rational Indices

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Example – 13

Find the sum of the series

\[1 + \frac{2}{6} + \frac{{2.5}}{{6.12}} + \frac{{2 \cdot 5 \cdot 8}}{{6 \cdot 12 \cdot 18}} + ......\infty \]

if you are told that this corresponds to an expansion of a binomial, of the form \({(1 + x)^n}\) .

Solution:    We need to determine n and x. For that, we can compare the terms of this series with the corresponding terms in the following general expansion.

\[{(1 + x)^n} = 1 + nx + \frac{{n(n - 1)}}{{2!}}{x^2} + \frac{{n(n - 1)\;(n - 2)}}{{3!}}{x^3} + ......\infty \]

Thus,

\[\begin{align}{} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,nx = \frac{2}{6}\\\\ \frac{{n(n - 1)}}{{2!}}{x^2} = \frac{{2.5}}{{6.12}} \end{align}\]

Solving for n and x from these two equations, we get \(\begin{align}n =  - \frac{2}{3}\end{align}\) and \(\begin{align}x =  - \frac{1}{2}\end{align}\) . Thus, the sum of the series is

\[\begin{array}{l}S = {\left( {1 + x} \right)^n} = {\left( {1 - \frac{1}{2}} \right)^{ - \frac{2}{3}}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {4^{1/3}}\end{array}\]

Example – 14

Find the sum of the series

\[\begin{align}\frac{1}{{a + b}} + \frac{1}{{a + 2b}} + \frac{1}{{a + 3b}} + ...... to\;\; n\;\; terms\end{align}\]

for b << a

Solution:    Before solving this problem, ponder a moment over the following fact:

In the expansion of \({(1 + x)^n}\) , if x << 1, that is, if x is much smaller than 1, then the expansion can be approximated as

\[{(1 + x)^n} \approx 1 + nx\]

since all higher order terms can be neglected due to the small magnitude of x.

Coming to the problem, note that if b << a, i.e, if \(\frac{b}{a} <  < 1\) , then,

\[\begin{align}{}\frac{1}{{a + rb}} &= \frac{1}{{a\left( {1 + r\frac{b}{a}} \right)}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,& = \frac{1}{a}{\left( {1 + r\frac{b}{a}} \right)^{ - 1}}\end{align}\]

\[ \approx \frac{1}{a}\left( {1 - \frac{{rb}}{a}} \right) since \;\;\frac{b}{a} <  < 1\]

Thus, the sum S of the series is (to a good approximation)

\[\begin{align}{}S &\approx \sum\limits_{r = 1}^n {} \frac{1}{a}\left( {1 - r\frac{b}{a}} \right)\\\\\,\,\, &= \frac{1}{a}\left( {n - \frac{{n(n + 1)}}{2}\frac{b}{a}} \right).\\\\\,\,\, &= \frac{n}{a} - \frac{{n(n + 1)b}}{{2{a^2}}}\end{align}\]

Example – 15

Evaluate 99 3/2  correct to four decimal places.

Solution:    We have

\[\begin{array}{l}{99^{3/2}}\;\; = {\left( {100 - 1} \right)^{3/2}}\\\\\,\,\,\,\,\,\,\,\,\,\,\, = {100^{3/2}} \cdot {\left( {1 - 0.01} \right)^{3/2}}\\\\\,\,\,\,\,\,\,\,\,\,\, = 1000 \cdot {\left( {1 - 0.01} \right)^{3/2}}\\\\\,\,\,\,\,\,\,\,\,\,\, = 1000 \cdot \left( {1 - \frac{3}{2}\; \cdot \;\left( {0.01} \right) + \frac{{\frac{3}{2}\; \cdot \;\frac{1}{2}}}{{2!}}\; \cdot \;{{\left( {0.01} \right)}^2} - ......} \right)\\\\\,\,\,\,\,\,\,\,\, = 1000\left( {1 - \frac{3}{{200}} + \frac{3}{{80000}} - .......} \right)\\\\\,\,\,\,\,\,\,\, = 1000 - 15 + 0.0375 - ......\\\\\,\,\,\,\,\,\, = {\rm{ }}985.0375\end{array}\]

Note that we only considered the first three terms of the expansion because the higher order terms would not have had any effect on the answer upto the fourth decimal place.

TRY YOURSELF - II

Q. 1      Find the sum of the series

\[1 + \frac{1}{3} + \frac{{1.3}}{{3.6}} + \frac{{1.3.5}}{{3.6.9}} + ......\infty \]

Q. 2      Find the sum of the series

\[1 + \frac{1}{4} + \frac{{1.3}}{{4.8}} + \frac{{1.3.5}}{{4.8.12}} + ......\infty \]

Q. 3      Find the magnitude of the greatest term in the expansion of \({\left( {1 + 3y} \right)^{ - 2/5}}\) for \(y = \begin{align}\frac{1}{5}\end{align}\) .

Q. 4    Find the magnitude of the greatest term in the expansion of \({\left( {1 - 5y} \right)^{3/5}}\) for \(y = \begin{align}\frac{1}{3}\end{align}\)

Q. 5      If \(|x|\; <  < 1\) , find the approximate value of

\[\frac{{\sqrt {1 + x} \; + \;\sqrt {{{(1 - x)}^3}} }}{{\sqrt {1 + x}  + \sqrt {{{(1 - x)}^5}} }}\]

Q. 6      If \(\,y = x - {x^2} + {x^3} - {x^4} + ......\infty \) , show using the general binomial theorem that

\(\begin{align}\qquad\qquad x = \frac{y}{{1 - y}}\end{align}\)

Q. 7 If a is very nearly equal to b, then show that the value of \(\begin{align}\frac{{b + 2a}}{{a + 2b}}\end{align}\) is nearly equal to \(\begin{align}{\left( {\frac{a}{b}} \right)^{\frac{1}{3}}}\end{align}\) .

Hint: Write \(\begin{align}\frac{a}{b} = 1 + x , \quad where\;\; |x|\; <  < 1\end{align}\)

Q. 8      Using the general binomial theorem, find the approximate value for 9 3/2 .

Q. 9 *    Prove that the coefficient of x n  in the expanded represent action of

\[\frac{{1 + {x^2} - {x^4}}}{{{{(1 + x)}^3}}}\]

will be equal to

\[{\left( { - 1} \right)^n}\left( {\frac{{{n^2} + 5n - 8}}{2}} \right)\]

Q. 10 Find the coefficient of x n  in the expanded representation of \(\begin{align}\frac{x}{{(x - a)\;(x - b)}} ,\; if\;\; |x|\; < min (a, b)\end{align}\) 

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