Finding Taylor series about some point

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SUMMARY

The discussion focuses on finding the Taylor series of sin(2x + 1) around the point x = 0. It establishes that direct substitution into the Maclaurin series for sin(x) is invalid because 2x + 1 does not approach 0 as x approaches 0. Instead, the correct approach is to derive the Taylor series for sin(x) centered at x = 1 and then substitute 2x + 1 for the variable to obtain a series in powers of 2x.

PREREQUISITES
  • Understanding of Taylor series and Maclaurin series
  • Familiarity with the function sin(x) and its properties
  • Knowledge of substitution techniques in series expansion
  • Basic calculus concepts, particularly limits and continuity
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  • Study the derivation of Taylor series for sin(x) around different points
  • Learn about the convergence of Taylor series and their radius of convergence
  • Explore substitution methods in series expansions with examples
  • Investigate the implications of Taylor series in approximating functions
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Students of calculus, mathematicians, and educators looking to deepen their understanding of Taylor series and their applications in function approximation.

JG89
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In this: http://www.math.tamu.edu/~fulling/coalweb/sinsubst.pdf


It says that to find the Taylor series of sin(2x + 1) around the point x = 0, we cannot just substitute 2x+1 into the Maclaurin series for sinx because 2x + 1 doesn't approach a limit of 0 as x approaches 0.

It says we have to find the Taylor series for sinx about x = 1, and then we can substitute 2x + 1 into it.

Why is this?
 
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The Taylor series for sinu around 1 will be a series of powers of (u-1). Substitute 2x+1 for u and you will have a series in powers of 2x.
 

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