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Mathematics
Calculus
Taylor series and variable substitutions
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[QUOTE="PeroK, post: 6311858, member: 493650"] One issue is that you are using ##f## to denote two different functions. If we say that ##f(x) = \frac{1}{1- x^2}##, and ##x = \sin \theta##, then we can define: $$g(\theta) = f(\sin \theta) = \frac{1}{1- \sin^2 \theta} = \sec^2 \theta$$ And you can see that ##g## is not the same function as ##f##. For example, where you have: This is wrong. By definition of the function ##f## we have: $$f(\theta) = \frac{1}{1-\theta^2} \approx 1+\theta^2+\theta^4+ \dots$$ The second Taylor series is actually for the composite function ##g##: $$g(\theta) = \sec^2\theta \approx 1+\theta^2+\frac{2}{3}\theta^4+ \dots $$ [/QUOTE]
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Forums
Mathematics
Calculus
Taylor series and variable substitutions
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