Maclaurin's Series: Expanding tan(ex) up to x2

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SUMMARY

The discussion focuses on expanding the function tan(ex) using Maclaurin's Series up to the x² term. Participants emphasize the importance of including all relevant terms that contribute to the coefficient of x² in the series. The consensus is that directly applying Maclaurin's Series to tan(ex) is the most effective approach to ensure accuracy in the expansion. The conversation highlights the necessity of careful term consideration to avoid discrepancies in results.

PREREQUISITES
  • Understanding of Maclaurin's Series
  • Familiarity with Taylor series expansions
  • Basic knowledge of the function tan(x)
  • Experience with exponential functions, specifically e^x
NEXT STEPS
  • Study the derivation of Maclaurin's Series for various functions
  • Learn how to compute Taylor series expansions for functions like tan(x)
  • Explore the properties of exponential functions and their series expansions
  • Practice finding coefficients in power series to enhance accuracy
USEFUL FOR

Students studying calculus, particularly those focusing on series expansions, mathematicians interested in power series, and educators teaching Maclaurin's Series applications.

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Homework Statement



Find all the terms up to and including x2 in the power series for tan(ex).

Homework Equations





The Attempt at a Solution



I wonder can i expand it as follow... why will i get different answer when I use Maclaurin's Series to expand it?

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Michael_Light said:

Homework Statement



Find all the terms up to and including x2 in the power series for tan(ex).

Homework Equations


The Attempt at a Solution



I wonder can i expand it as follow... why will i get different answer when I use Maclaurin's Series to expand it?

View attachment 43613

Well, what's the next term (the one with e5x)? Does it contribute a term with x2? How about the one after that? ...etc.

If you're asked to find a series sum up to and including the x2 term, you have to be certain that you've accounted for ALL the terms that can contribute to the coefficient of x2 in the final series.

You're better off just applying Maclaurin's directly to the function tan(ex).
 

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