Maclaurin Series and Power Series - Convergence and Validity of Power Series
Maclaurin Series and Power Series - Convergence and Validity of Power Series
TGM Original Questions
Find the first four terms in the series expansion of \({{(4+{{x}^{2}})}^{\frac{1}{2}}}\)
- ascending powers of \(x\)
- 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) \(\displaystyle \sqrt{4+x^2}=2+\frac{x^2}{4}-\frac{x^4}{64}+\frac{x^6}{512}+O(x^8)\), valid for \(-2<x<2\). (ii) For \(x>2\), \(\displaystyle \sqrt{4+x^2}=x+\frac{2}{x}-\frac{2}{x^3}+\frac{4}{x^5}+O(x^{-7})\). For \(x<-2\), \(\displaystyle \sqrt{4+x^2}=-x-\frac{2}{x}+\frac{2}{x^3}-\frac{4}{x^5}+O(|x|^{-7})\). Equivalently, \(\displaystyle \sqrt{4+x^2}=|x|+\frac{2}{|x|}-\frac{2}{|x|^3}+\frac{4}{|x|^5}+O(|x|^{-7})\), valid for \(|x|>2\).
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