Taylor Series: Step-by-Step Guide
A Taylor series represents a smooth function as an infinite polynomial built from its derivatives at a point a. When a = 0 it is called a Maclaurin series. Within its interval of convergence, the series equals the function exactly; truncating gives polynomial approximations.
Math Racer skill guide · Calculus 2 · AP Calc BC / College
How to solve it, step by step
- Choose the center a. Pick the point about which you expand. a = 0 (Maclaurin) is most common; sometimes a = 1 or another nearby point is more natural.\[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n\]
- Compute derivatives at a. Differentiate f repeatedly and evaluate each derivative at x = a until you spot a pattern.\[f(a),\; f'(a),\; f''(a),\; f'''(a),\;\dots\]
- Write a general formula for the nth term. Express f^(n)(a)/n! in closed form, then assemble the series.\[c_n = \frac{f^{(n)}(a)}{n!}\]
- Use known series when possible. Substitute, multiply, differentiate, or integrate known Maclaurin series for e^x, sin x, cos x, 1/(1-x), and ln(1+x) instead of computing derivatives.\[e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}\]
- State the interval of convergence. Determine where the series converges to f. For e^x, sin x, cos x this is all real x; for 1/(1-x) it is |x| < 1.\[|x - a| < R\]
Worked example
Find the Maclaurin series for f(x) = e^(2x) and its radius of convergence.
Step 1: Center at a = 0 (Maclaurin series).
Step 2: Use the known series e^u = sum u^n / n! and substitute u = 2x.
Step 3: e^(2x) = sum from n=0 to infinity of (2x)^n / n! = sum 2^n x^n / n!.
Step 4: Confirm by computing derivatives at 0: f^(n)(x) = 2^n e^(2x), so f^(n)(0) = 2^n. Thus c_n = 2^n / n!.
Step 5: The series sum 2^n x^n / n! converges for all real x (the substitution preserves the infinite radius of convergence of e^u).
Verification: Truncating at n=2 gives 1 + 2x + 2x^2, which agrees with the second-degree Taylor polynomial of e^(2x) at 0.
Tips and common mistakes
- Memorize the Maclaurin series for e^x, sin x, cos x, 1/(1-x), and ln(1+x).
- Substitute, differentiate, integrate, multiply, or add known series instead of computing derivatives by hand.
- Always check the interval of convergence after manipulation; substitution may shrink or expand it.
- f^(n)(a)/n! is the formula; don't forget to divide by n!.
- For Maclaurin series, look for symmetry: even functions have only even powers; odd functions have only odd powers.
- The Taylor remainder bounds the error of truncation: use Lagrange's form for explicit estimates.
- Differentiating a Taylor series term by term gives the Taylor series of the derivative.
- Replace x with -x, x^2, or any expression to derive new series quickly.