Maclaurin Series and Power Series - Binomial Expansion as a Maclaurin Series
Maclaurin Series and Power Series - Binomial Expansion as a Maclaurin Series
Obtain the first four terms in the expansion of \({{(1+3x)}^{-1}}\).
- in ascending powers of \(x\),
- in descending powers of \(x\).
For each case, state the range of values of \(x\) for which the expansion is valid.
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Answer:(i) \(\displaystyle (1+3x)^{-1}=1-3x+9x^2-27x^3+O(x^4)\), valid for \(\displaystyle -\frac13<x<\frac13\). (ii) \(\displaystyle (1+3x)^{-1}=\frac{1}{3x}-\frac{1}{9x^2}+\frac{1}{27x^3}-\frac{1}{81x^4}+O(x^{-5})\), valid for \(\displaystyle x< -\frac13\) or \(\displaystyle x>\frac13\).
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