Partial Fractions - An ounce of heart . . . A spoonful of education

by Mr Will - April 27, 2019December 2, 2022

Binomial Expansion and Binomial Series

In algebra, we all have learnt the following basic algebraic expansion:

\( \hspace{3em} {(a + b)}^{2} = {a}^{2} + 2ab + {b}^{2} \).

We can keep multiplying the expression \( { \small (a + b) } \) by itself to find the expression for higher index value. For example:

\(\\[12pt] {(a + b)}^{3} \ = \ {(a + b)}^{2} \times (a + b) \)
\(\\[12pt] \hspace{1.5em} \ = \ ({a}^{2} + 2ab + {b}^{2}) \times (a + b) \)
\(\\[12pt] \hspace{1.5em} \ = \ {a}^{3} + {a}^{2}b + 2{a}^2b + 2a{b}^2 + a{b}^2 + {b}^{3} \)
\( \hspace{1.5em} \ = \ {a}^{3} + 3{a}^{2}b + 3a{b}^2 + {b}^{3} \)

Of course this is a really tedious way for a very large index/power/exponent number.

Instead of doing it manually, we could use a formula called the Binomial Theorem which is shown below:

\( {(a + b)}^{n} \ = \ \displaystyle \sum_{k=0}^{n} \binom{n}{k} {a}^{n \ – \ k} \ {b}^{k} \)

with \( \displaystyle \binom{n}{k} \ = \ \frac{n!}{k!(n-k)!} \)

\(\hspace{4.4em} = \ {\large \frac{n (n \ – \ 1) (n \ – \ 2) … (n \ – \ k \ + \ 1)}{k(k \ – \ 1)(k \ – \ 2) \ … \ 1} } \)

for \( k \geq 1 \) and \( \displaystyle \binom{n}{0} \ = \ 1 \)

\(\\[10pt]\) Example:
\(\\[12pt] {(a + b)}^{10} \)
\(\\[15pt] = \binom{10}{0} {a}^{10}{b}^{0} + \binom{10}{1} {a}^{9}{b}^{1}+…+ \binom{10}{10} {a}^{0}{b}^{10} \)
\(\\[15pt] = {a}^{10} + 10 {a}^{9}b+ 45 {a}^{8}{b}^{2} + … + 10{a}{b}^{9} + {b}^{10} \)

\(\\[10pt]\) Note that,
\(\\[20pt]\quad \ {\displaystyle \binom{10}{0} } \ = \ 1 \)
\(\\[20pt]\quad \ {\displaystyle \binom{10}{1} } \ = \ {\large \frac{ 10 }{ 1 } } \ = \ 10 \)
\(\\[20pt]\quad \ {\displaystyle \binom{10}{2} } \ = \ {\large \frac{ 10 \ \times \ 9 }{ 2 \ \times \ 1 } } \ = \ 45 \)
\(\\[20pt]\quad \ {\displaystyle \binom{10}{3} } \ = \ {\large \frac{ 10 \ \times \ 9 \ \times \ 8 }{ 3 \ \times \ 2 \ \times \ 1 } } \ = \ 120 \)
\(\\[20pt]\quad \ {\displaystyle \binom{10}{4} } \ = \ {\large \frac{ 10 \ \times \ 9 \ \times \ 8 \ \times \ 7 }{4 \ \times \ 3 \ \times \ 2 \ \times \ 1 } } \ = \ 210 \)
\(\\[20pt]\quad \ {\displaystyle \binom{10}{5} } \ = \ {\large \frac{10 \ \times \ 9 \ \times \ 8 \ \times \ 7 \ \times \ 6 }{5 \ \times \ 4 \ \times \ 3 \ \times \ 2 \ \times \ 1 } } \ = \ 252 \)
, etc.

\(\binom{n}{k} \) is read as “n choose k” or sometimes referred to as the binomial coefficients.

Notice that this binomial expansion has a finite number of terms with the k values take the non-negative numbers from 0, 1, 2, … , n.

Then the next question would be: Can we still use the binomial theorem for the expansion with negative number or fractional number for the index value?

Thankfully, Sir Isaac Newton has shown that the binomial theorem can be generalized to take in any numbers for the index value including the negative and fractional numbers as long as it is within a convergence rule.

Using this result, we have the Binomial Series which can be expressed as follows:

\(\\[25pt]{(1 + x)}^{n} = \displaystyle \sum_{k=0}^{\infty} \binom{n}{k} {x}^{k} \)
\(\\[10pt]{\small = \ 1 + nx + \frac{(n)(n-1)}{2!}{x}^{2} + \frac{(n)(n-1)(n-2)}{3!}{x}^{3} + … }\)

with \( {\small k } \) is any real number and \( {\small -1 \lt x \lt 1 } \).

\(\\[10pt]\) Example:
\(\\[12pt] { {\large \frac{1}{1 + x}} \ = \ (1 + x)}^{-1} \)
\(\\[15pt]{\small = \ 1 + (-1)x + \frac{(-1)(-2)}{2!}{x}^{2} + \frac{(-1)(-2)(-3)}{3!}{x}^{3} + … }\)
\(\\[15pt]{\small = \ 1 \ – \ x \ + \ {x}^{2} \ – \ {x}^{3} \ + \ … \ – \ … }\)
\(\\[15pt]\) The expansion is valid when \({\small -1 \lt x \lt 1 } \).

\(\\[10pt]\) Another example:
\(\\[12pt] { \frac{1}{\sqrt{(1 + x)}} \ = \ (1 + x)}^{\frac{1}{2}} \)
\(\\[15pt]{\small = \ 1 + \frac{1}{2}x + \frac{(\frac{1}{2})(- \frac{1}{2})}{2!}{x}^{2} + \frac{(\frac{1}{2})(- \frac{1}{2})( – \frac{3}{2})}{3!}{x}^{3} + … }\)
\(\\[15pt]{ = \ 1 + \frac{x}{2} – \frac{{x}^{2}}{8} + \frac{{x}^{3}}{16} \ – \ … + … }\)
\(\\[15pt]\) The expansion is valid when \({\small -1 \lt x \lt 1 } \).

Try some of the examples below and if you need any help, just look at the solution I have written. Cheers ! =) .
\(\\[1pt]\)

EXAMPLE:

\({\small 1.\enspace}\) Let \({\small f(x) \ = \ {\large \frac{7{x}^{2} \ – \ 15x \ + \ 8}{(1 \ – \ 2x){(2 \ – \ x)}^{2} }} }\)
\(\\[1pt]\)
Express \({\small f(x) }\) in partial fractions and hence obtain the expansion of \({\small f(x) }\) in ascending powers of x, up to and including the term in \({\small {x}^{2}}\).