A Maclaurin series is a power series that represents a function using powers of x centered at x = 0. These series matter because they turn complicated functions like e^x, sin x, and cos x into polynomials that are easier to calculate, graph, and analyze. In calculus, they connect derivatives, approximation, and infinite sums in one powerful idea.
They are also used throughout physics and engineering to model motion, waves, signals, and small changes.
Key Facts
- Maclaurin series formula: f(x) = f(0) + f'(0)x + f''(0)x^2/2! + f'''(0)x^3/3! + ...
- Exponential series: e^x = 1 + x + x^2/2! + x^3/3! + ... = sum from n = 0 to infinity of x^n/n!
- Sine series: sin x = x - x^3/3! + x^5/5! - x^7/7! + ...
- Cosine series: cos x = 1 - x^2/2! + x^4/4! - x^6/6! + ...
- Geometric series: 1/(1 - x) = 1 + x + x^2 + x^3 + ... for |x| < 1
- New series can be built by substitution, differentiation, integration, and multiplication of known Maclaurin series.
Vocabulary
- Maclaurin series
- A Maclaurin series is a Taylor series centered at x = 0 that writes a function as an infinite polynomial.
- Power series
- A power series is an infinite sum of terms involving powers of a variable, usually written in the form sum a_n x^n.
- Coefficient
- A coefficient is the number multiplying a power of x in a polynomial or power series.
- Radius of convergence
- The radius of convergence is the distance from the center of a power series within which the series converges.
- Partial sum
- A partial sum is a polynomial formed by adding only the first several terms of an infinite series.
Common Mistakes to Avoid
- Forgetting factorials in e^x, sin x, and cos x. The denominators 2!, 3!, 4!, and so on are essential because they come from the derivative formula for Taylor coefficients.
- Using the geometric series outside its convergence interval. The formula 1/(1 - x) = 1 + x + x^2 + ... is valid only when |x| < 1.
- Mixing up the sine and cosine patterns. Sine has only odd powers and starts with x, while cosine has only even powers and starts with 1.
- Substituting incorrectly when building a new series. If x is replaced by 2x or x^2, every occurrence of x in the original series must be replaced, including the powers.
Practice Questions
- 1 Use the first four nonzero terms of the Maclaurin series for e^x to approximate e^0.2.
- 2 Find the Maclaurin series for sin(3x) through the x^7 term.
- 3 Explain why the Maclaurin series for cos x contains only even powers of x, while the Maclaurin series for sin x contains only odd powers of x.