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Topic 2 of 6

Common Maclaurin Series

Six Maclaurin series appear constantly in physics. Knowing them cold — and being able to quickly identify which one applies — is one of the highest-leverage skills in theoretical physics.

Key Concepts

Exponential exe^x
ex=1+x+x2/2!+x3/3!+⋯e^x = 1 + x + x^2/2! + x^3/3! + \cdots for all xx. Appears in quantum mechanics, statistical mechanics, decay laws, and wave functions.
Trigonometric: sin⁡x\sin x, cos⁡x\cos x
sin⁡x=x−x3/3!+x5/5!−⋯\sin x = x - x^3/3! + x^5/5! - \cdots and cos⁡x=1−x2/2!+x4/4!−⋯\cos x = 1 - x^2/2! + x^4/4! - \cdots, both for all xx. Foundation of the small-angle approximation.
Logarithm: ln⁡(1+x)\ln(1+x)
ln⁡(1+x)=x−x2/2+x3/3−⋯\ln(1+x) = x - x^2/2 + x^3/3 - \cdots for ∣x∣≤1|x| \leq 1. Appears in entropy, information theory, and any Taylor expansion near unity.
Binomial: (1+x)α(1+x)^\alpha
(1+x)α=1+αx+α(α−1)2!x2+⋯(1+x)^\alpha = 1 + \alpha x + \frac{\alpha(\alpha-1)}{2!}x^2 + \cdots for ∣x∣<1|x| < 1. Includes 1+x\sqrt{1+x}, 1/1−β21/\sqrt{1-\beta^2} (Lorentz factor), and all fractional powers.
Geometric: 1/(1−x)1/(1-x)
1/(1−x)=1+x+x2+x3+⋯1/(1-x) = 1 + x + x^2 + x^3 + \cdots for ∣x∣<1|x| < 1. The simplest power series; a special case of the binomial with α=−1\alpha = -1.

Key Equations

exe^x
ex=∑n=0∞xnn!=1+x+x22!+x33!+⋯e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots
Converges for all xx.
sin⁡x\sin x and cos⁡x\cos x
sin⁡x=∑n=0∞(−1)nx2n+1(2n+1)!,cos⁡x=∑n=0∞(−1)nx2n(2n)!\sin x = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{(2n+1)!}, \quad \cos x = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{(2n)!}
Both converge for all xx.
Binomial Series
(1+x)α=∑n=0∞(αn)xn=1+αx+α(α−1)2!x2+⋯(1+x)^\alpha = \sum_{n=0}^{\infty}\binom{\alpha}{n}x^n = 1 + \alpha x + \frac{\alpha(\alpha-1)}{2!}x^2 + \cdots
Converges for ∣x∣<1|x| < 1 (any α\alpha) and for all xx when α\alpha is a non-negative integer.
Worked Example

Computing the Maclaurin Series of cos⁡(x)\cos(x)

Problem

Derive the Maclaurin series for cos⁡(x)\cos(x) by repeatedly differentiating.

Solution

The derivatives of cos⁡(x)\cos(x) cycle with period 4: f(0)=cos⁡xf^{(0)}=\cos x, f(1)=−sin⁡xf^{(1)}=-\sin x, f(2)=−cos⁡xf^{(2)}=-\cos x, f(3)=sin⁡xf^{(3)}=\sin x, f(4)=cos⁡x,…f^{(4)}=\cos x, \ldots

Evaluating at x=0x=0: f(0)=1f(0)=1, f′(0)=0f'(0)=0, f′′(0)=−1f''(0)=-1, f′′′(0)=0f'''(0)=0, f(4)(0)=1f^{(4)}(0)=1, etc. Only even-order terms survive.

cos⁡x=1−x22!+x44!−x66!+⋯=∑n=0∞(−1)nx2n(2n)!\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{(2n)!}

The alternating signs come from the pattern +1,−1,+1,…+1, -1, +1, \ldots of the even-order derivatives at x=0x=0.

Answer cos⁡x=1−x2/2!+x4/4!−x6/6!+⋯\cos x = 1 - x^2/2! + x^4/4! - x^6/6! + \cdots, converging for all xx

Interactive: Series Visualizer

Adjust the number of terms and watch the Taylor approximation converge to the exact function. Blue is exact; yellow is the truncated series.

16
Current approximation
T3(x)=x  −x36  +x5120T_{3}(x) = x \; -\dfrac{x^3}{6} \; +\dfrac{x^5}{120}
Exact f(x)
T3(x)

For sin, cos, and eˣ the series converges everywhere. For ln(1+x) convergence is limited to |x| < 1 — watch the approximation diverge outside that interval.

Practice

Exercises

6 problems
1 of 6

Using the 2-term series sin⁡x≈x−x3/6\sin x \approx x - x^3/6, compute sin⁡(0.3)\sin(0.3). Give to 4 decimal places.

2 of 6

Using the 4-term series ex≈1+x+x2/2+x3/6e^x \approx 1 + x + x^2/2 + x^3/6, compute e0.5e^{0.5}. Give to 4 decimal places.

3 of 6

Using the leading binomial approximation (1+x)1/2≈1+x/2(1+x)^{1/2} \approx 1 + x/2 for small xx, approximate 1.06\sqrt{1.06}.

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4 of 6

Using the 3-term series cos⁡x≈1−x2/2+x4/24\cos x \approx 1 - x^2/2 + x^4/24, compute cos⁡(1)\cos(1). Give to 4 decimal places.

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5 of 6

Using the 4-term geometric series 1/(1+x)≈1−x+x2−x31/(1+x) \approx 1 - x + x^2 - x^3 with x=0.2x = 0.2, approximate 1/1.21/1.2.

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6 of 6

Using the 3-term series ln⁡(1+x)≈x−x2/2+x3/3\ln(1+x) \approx x - x^2/2 + x^3/3 with x=0.1x = 0.1, approximate ln⁡(1.1)\ln(1.1). Give to 5 decimal places.

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Key Takeaways

  • The six essential series: exe^x, sin⁡x\sin x, cos⁡x\cos x, ln⁡(1+x)\ln(1+x), (1+x)α(1+x)^\alpha, and 1/(1−x)1/(1-x).
  • exe^x, sin⁡x\sin x, and cos⁡x\cos x converge for all real xx. The others converge only for ∣x∣<1|x| < 1 (or ∣x∣≤1|x| \leq 1 in some cases).
  • The binomial series (1+x)α=1+αx+…(1+x)^\alpha = 1 + \alpha x + \ldots is perhaps the most useful — it covers square roots, reciprocals, and the Lorentz factor.
  • Euler's formula eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos\theta + i\sin\theta follows directly from the exponential and trig series.
  • When you see a complicated expression involving these functions with small arguments, reach for the series immediately.
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