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Mathematics · University

A Taylor coefficient of an exponential

The requested coefficient is not the nth derivative; divide by n factorial.

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Method and conditions

The requested coefficient is not the nth derivative; divide by n factorial.

[xⁿ] e^(kx) = kⁿ/n!

Worked examples

Find the coefficient of x^3 in the Maclaurin series of 4e^(2x). Round the final result to 4 decimal places if needed.

Use e^(2x) = Σ (2x)^j/j!. Multiply the j = 3 term by 4.

5.3333

Find the coefficient of x^6 in the Maclaurin series of 5e^(7x). Round the final result to 4 decimal places if needed.

Use e^(7x) = Σ (7x)^j/j!. Multiply the j = 6 term by 5.

817.0069

Try it

  1. Find the coefficient of x^5 in the Maclaurin series of 5e^(6x). Round the final result to 4 decimal places if needed.

  2. Find the coefficient of x^7 in the Maclaurin series of 5e^(3x). Round the final result to 4 decimal places if needed.

Check your work

Answers and marking notes
  1. 324

    Use e^(6x) = Σ (6x)^j/j!. Multiply the j = 5 term by 5.

  2. 2.1696

    Use e^(3x) = Σ (3x)^j/j!. Multiply the j = 7 term by 5.

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