Aim for FiveCalculus BC

Unit 10 · Topic 10.14

BC only

10.14 Finding Taylor or Maclaurin Series for a Function

BC only. A Taylor series is the never-ending version of a Taylor polynomial: Σ f⁽ⁿ⁾(a)(x − a)ⁿ/n!. Know the Maclaurin series for eˣ, sin x, cos x and 1/(1 − x) by heart, because many other series are built from them.

Key terms

  • Taylor series
  • Maclaurin series
  • general term
  • series for eˣ, sin x, cos x and 1/(1 − x)

From polynomial to series

This whole unit is BC only. The Taylor series for f centered at x = a is

Σ (n = 0 to ∞) f⁽ⁿ⁾(a)(x − a)ⁿ/n! = f(a) + f′(a)(x − a) + f″(a)(x − a)²/2! + … .

Each Taylor polynomial Pₙ is a partial sum of this series. Centered at 0, it's the Maclaurin series. Where the series converges to f(x), you can treat f(x) as an infinitely long polynomial.

The four series to memorize

Patterns that help you remember: sin x is odd and uses odd powers; cos x is even and uses even powers; both alternate signs. The series for eˣ uses every power with all positive signs. The series for 1/(1 − x) is the geometric series (10.2) with first term 1 and ratio x.

FunctionMaclaurin seriesGeneral termConverges for
eˣ1 + x + x²/2! + x³/3! + …xⁿ/n!all real x
sin xx − x³/3! + x⁵/5! − …(−1)ⁿ x²ⁿ⁺¹/(2n + 1)!all real x
cos x1 − x²/2! + x⁴/4! − …(−1)ⁿ x²ⁿ/(2n)!all real x
1/(1 − x)1 + x + x² + x³ + …xⁿ−1 < x < 1

Finding a general term

  • Write the first several terms.
  • Spot the pattern in the powers (all, odd, even), the signs (constant or alternating) and the coefficients (factorials, integers).
  • Use (−1)ⁿ for alternating signs starting with +, or (−1)ⁿ⁺¹ starting with −.
  • Check your general term against the first two or three terms by plugging in n = 0, 1, 2.

Using series

Series let you evaluate hard limits, approximate integrals and identify functions. For limits, substitute the series and simplify: (sin x − x)/x³ becomes (−x³/6 + x⁵/120 − …)/x³ = −1/6 + x²/120 − …, which approaches −1/6 as x → 0.

A Taylor series doesn't automatically equal its function everywhere; it can only equal f where it converges. For the four basic series, they equal their functions on the whole interval shown in the table.

If you're handed a series and asked what function it represents, compare it with the four above. Σ (−1)ⁿ x²ⁿ/(2n)! is cos x, and Σ xⁿ/n! is eˣ.

Worked examples

Try each one yourself first, then open the solution.

  1. Example 1

    Deriving the cos x series

    Find the Maclaurin series for cos x, including the general term.

    Show the solution
    1. Step 1: Derivatives cycle: cos x, −sin x, −cos x, sin x, cos x, … .
    2. Step 2: At 0: 1, 0, −1, 0, 1, 0, −1, … .
    3. Step 3: Nonzero terms: 1 − x²/2! + x⁴/4! − x⁶/6! + … .
    4. Step 4: Only even powers appear, with alternating signs: the general term is (−1)ⁿ x²ⁿ/(2n)!, for n = 0, 1, 2, … .
    5. Step 5: Check n = 1: (−1)x²/2! = −x²/2. ✓

    Answer: cos x = Σ (n = 0 to ∞) (−1)ⁿ x²ⁿ/(2n)! = 1 − x²/2! + x⁴/4! − …

  2. Example 2

    A Taylor series not centered at 0

    Find the Taylor series for ln x centered at x = 1.

    Show the solution
    1. Step 1: f = ln x, f′ = 1/x, f″ = −1/x², f‴ = 2/x³, f⁽⁴⁾ = −6/x⁴.
    2. Step 2: At x = 1: 0, 1, −1, 2, −6. In general, f⁽ⁿ⁾(1) = (−1)ⁿ⁺¹(n − 1)! for n ≥ 1.
    3. Step 3: Divide by n!: (−1)ⁿ⁺¹(n − 1)!/n! = (−1)ⁿ⁺¹/n.
    4. Step 4: ln x = (x − 1) − (x − 1)²/2 + (x − 1)³/3 − … = Σ (n = 1 to ∞) (−1)ⁿ⁺¹(x − 1)ⁿ/n.

    Answer: ln x = Σ (n = 1 to ∞) (−1)ⁿ⁺¹(x − 1)ⁿ/n

  3. Example 3

    Trap: using a series for a limit

    Find lim (x → 0) (sin x − x)/x³.

    Show the solution
    1. Step 1: Substitute the series: sin x − x = (x − x³/6 + x⁵/120 − …) − x = −x³/6 + x⁵/120 − … .
    2. Step 2: Divide by x³: −1/6 + x²/120 − … .
    3. Step 3: As x → 0, every term after the first goes to 0.
    4. Step 4: Trap: writing sin x ≈ x and getting 0/0 = 0. You need enough terms to see what's left after the cancellation.

    Answer: −1/6

Common mistakes

  • Mixing up the sin and cos series (odd vs. even powers, (2n + 1)! vs. (2n)!).
  • Forgetting the alternating signs in sin and cos.
  • Writing a general term that doesn't match the first terms. Always test n = 0 and n = 1.
  • Treating 1/(1 − x) as valid for all x. It only converges for |x| < 1.

On the exam

  • BC free response frequently asks for the first four nonzero terms and the general term of a Taylor series. Show the derivatives (or the known series you start from).
  • Multiple-choice questions often show a series and ask what function it equals at a particular x, like Σ (−1)ⁿ π²ⁿ/(2n)! = cos π = −1.

Connected topics

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Check yourself

4 questions on 10.14 Finding Taylor or Maclaurin Series for a Function. Pick an answer to see if you got it, and why.

Question 1 of 4

What is the coefficient of x⁸ in the Maclaurin series for f(x) = x²e^(−x²)?

Question 2 of 4

What is the sum of the series Σ (n = 0 to ∞) (−1)ⁿ (π/3)²ⁿ⁺¹/(2n + 1)!?

Question 3 of 4

Which of the following is the Taylor series for f(x) = eˣ about x = 1?

The function f is defined by the power series f(x) = Σ (n = 1 to ∞) (x − 3)ⁿ/(n · 2ⁿ) = (x − 3)/2 + (x − 3)²/8 + (x − 3)³/24 + ⋯ for all x for which the series converges.

Described function

Question 4 of 4

What is the value of f‴(3)?

0 of 4 answered