Finding Taylor Series - different Method

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SUMMARY

The discussion focuses on finding the Taylor series for the function f(x) = sin(2x)ln(1-x) up to n = 4. A user expresses frustration with the lengthy process of taking derivatives to obtain the series. A solution is proposed to simplify the task by separately writing out the Taylor polynomials for sin(2x) and ln(1-x) and then multiplying them, avoiding the need for extensive differentiation.

PREREQUISITES
  • Understanding of Taylor series expansion
  • Familiarity with the functions sin(2x) and ln(1-x)
  • Basic calculus, specifically differentiation
  • Knowledge of polynomial multiplication
NEXT STEPS
  • Study the Taylor series for sin(x) and ln(1-x) in detail
  • Learn about polynomial multiplication techniques
  • Explore alternative methods for series expansion, such as using generating functions
  • Practice deriving Taylor series for other functions to reinforce understanding
USEFUL FOR

Students in calculus courses, mathematics enthusiasts, and anyone looking to simplify the process of finding Taylor series for complex functions.

JaeKyung
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Homework Statement



Hello, I'm in the middle of solving for the Taylor series of the function:

f(x)=sin(2x)ln(1-x)

up to n = 4.

The Attempt at a Solution



So far, I've been strictly taking its derivatives until I reach the fourth.
It's becoming a very long process considering it's just one of the many homework problems, and I can't help but think if there's a more elegant way of doing this.

Thank you in advance.
 
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Since you are only required to do this for n up to 4, I would suggest that you write out the Taylor's polynomials for sin(2x) and ln(1- x) separately (you should be familiar enough with those that you don't need to actually do the derivatives) and then multiply.
 

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