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Section 10.18 : Binomial Series
3. Write down the first four terms in the binomial series for \({\left( {1 + 3x} \right)^{ - 6}}\).
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Start SolutionFirst, we need to make sure it is in the proper form to use the Binomial Series from the notes which in this case we are already in the proper form with \(k = - 6\).
Show Step 2Now all we need to do is plug into the formula from the notes and write down the first four terms.
\[\begin{align*}{\left( {1 + 3x} \right)^{ - 6}} & = \sum\limits_{i = 0}^\infty { {-6 \choose i} {{\left( {3x} \right)}^i}} \\ & = 1 + \left( { - 6} \right){\left( {3x} \right)^1} + \frac{{\left( { - 6} \right)\left( { - 7} \right)}}{{2!}}{\left( {3x} \right)^2} + \frac{{\left( { - 6} \right)\left( { - 7} \right)\left( { - 8} \right)}}{{3!}}{\left( {3x} \right)^3} + \cdots \\ & = \require{bbox} \bbox[2pt,border:1px solid black]{{1 - 18x + 189{x^2} - 1512{x^3} + \cdots }}\end{align*}\]