Maclaurin Series and Power Series - Maclaurin Series Using Standard Series
Maclaurin Series and Power Series - Maclaurin Series Using Standard Series
TGM Original Questions
By using the standard series, find the following expansion up to and including the term \({{x}^{4}}\).
- \(\ln \left( 2+x \right)\)
- \(\ln \left( 5-x \right)\)
- \({{e}^{1-x}}\)
- \({{e}^{2+x}}\)
- \(\sin \left( 2x+1 \right)\)
- \(\cos \left( \frac{\pi }{3}-x \right)\)
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Answer:(a) \(\displaystyle \ln(2+x)=\ln2+\frac{x}{2}-\frac{x^2}{8}+\frac{x^3}{24}-\frac{x^4}{64}+O(x^5)\), \(-2<x\le2\). (b) \(\displaystyle \ln(5-x)=\ln5-\frac{x}{5}-\frac{x^2}{50}-\frac{x^3}{375}-\frac{x^4}{2500}+O(x^5)\), \(-5\le x<5\). (c) \(\displaystyle e^{1-x}=e\left(1-x+\frac{x^2}{2}-\frac{x^3}{6}+\frac{x^4}{24}+O(x^5)\right)\). (d) \(\displaystyle e^{2+x}=e^2\left(1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+O(x^5)\right)\). (e) \(\displaystyle \sin(2x+1)=\sin1+2x\cos1-2x^2\sin1-\frac{4x^3}{3}\cos1+\frac{2x^4}{3}\sin1+O(x^5)\). (f) \(\displaystyle \cos\left(\frac{\pi}{3}-x\right)=\frac12+\frac{\sqrt3}{2}x-\frac{x^2}{4}-\frac{\sqrt3}{12}x^3+\frac{x^4}{48}+O(x^5)\).
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