Math for ME · Chapter 6 of 19 · Intermediate
Sequences, Series, and Taylor Approximation
Every "small-angle" assumption, every linearised model, and most of numerical analysis is one idea: replace a hard function, near a point, with a short polynomial you can actually compute.
The thread: Calculus gives exact answers for clean functions. For the messy ones, this chapter trades exactness for a short polynomial you can actually compute, which is also the reason small-angle works.
Readiness check
From Derivatives and Trigonometry. Tick only what you can do closed-notes.
- Differentiate polynomials, ex, sin, and cos repeatedly.
- Evaluate a factorial such as 5! = 120.
- Use radians for trigonometry.
- Read Δy ≈ (dy/dx)Δx as a linear approximation from Derivatives.
- Compute percent error between two numbers.
The core idea
Near a point, any smooth function is almost a polynomial. The Taylor series is that polynomial, written out term by term.
f(x) ≈ f(a) + f′(a)(x − a) + f″(a)(x − a)²/2! + …Each term adds a derivative's worth of detail. Keep one term and you have the tangent line (Derivatives's linearisation); keep two and you capture curvature. A separate idea, the infinite series, sums infinitely many terms to a finite total when the terms shrink fast enough, which is what makes the approximation trustworthy.
The skills, taught in order
6.1 Sequences, series, and convergence
A sequence is an ordered list of numbers; a series is their running sum. The series converges if those partial sums approach a finite limit. A necessary first check is that the terms themselves shrink to zero, but that alone is not enough: the harmonic series 1 + 1/2 + 1/3 + … has terms going to zero yet sums to infinity.
6.2 The geometric series
One series has a clean closed form and appears everywhere, from settling sums to repeated reflections:
1 + r + r² + r³ + … = 1/(1 − r), for |r| < 1If the ratio r has magnitude below 1 the sum is finite; at or above 1 it diverges. This is the model for any process that adds a constant fraction of what remains.
6.3 Taylor and Maclaurin series
The Taylor series builds a polynomial from a function's derivatives at a point a; centred at a = 0 it is called a Maclaurin series. The four expansions worth memorising:
| Function | Series near 0 |
|---|---|
| ex | 1 + x + x²/2! + x³/3! + … |
| sin x | x − x³/3! + x⁵/5! − … |
| cos x | 1 − x²/2! + x⁴/4! − … |
| (1 + x)n | 1 + nx + n(n−1)x²/2! + … |
6.4 The approximations engineers actually use
Keep only the first term or two and the famous engineering shortcuts fall out:
sin θ ≈ θcos θ ≈ 1 − θ²/2(1 + x)n ≈ 1 + nxThe small-angle rule sin θ ≈ θ (θ in radians) linearises the pendulum and every vibration model. The binomial form (1 + x)n ≈ 1 + nx is the basis of sensitivity estimates: a small fractional change x, scaled by the exponent n.
6.5 How far can you trust it?
A Taylor series is exact at the point and degrades as you move away. The ratio test gives the radius of convergence, the distance over which the infinite series adds up at all, and the size of the first dropped term estimates the error of a truncated one. The practical rule: state how many terms you kept and over what range, then check the leftover term is small there.
Engineering connection: Dynamics and Vibrations (small-angle), Numerical Methods, Controls, and every linearised model.
Worked example: estimate e0.2 by hand
Use the Maclaurin series of ex to estimate e0.2 with three terms, then judge the error against the true value 1.2214.
- ProblemEstimate e0.2 from its series and find the truncation error.
- Given / findex = 1 + x + x²/2! + x³/3! + …, at x = 0.2. Find the three-term estimate and its error.
- ModelKeep terms up to x²; the first dropped term, x³/6, estimates the error.
- Equationex ≈ 1 + x + x²/2
- Compute1 + 0.2 + (0.2)²/2 = 1 + 0.2 + 0.02 = 1.220.
- Estimate the errorthe next term is x³/6 = (0.008)/6 = 0.0013, so the estimate should be low by about that much.
- Checktrue value 1.2214, so the actual error is 0.0014 (0.11%), matching the dropped-term estimate almost exactly. Adding that term gives 1.2213, within 0.01% of the truth.
- ConclusionThree terms of a Taylor series turned a transcendental function into school arithmetic, with a built-in error bar from the first dropped term. This is precisely how calculators, and every numerical method, evaluate ex, sin, and cos underneath.
Worked example 2: why the pendulum equation is allowed to be linear
A pendulum obeys the nonlinear equation θ″ + (g/L) sin θ = 0. Textbooks replace it with the linear θ″ + (g/L)θ = 0. Use the Taylor series to justify that step and find the angle at which the approximation reaches a 2% error.
- ProblemJustify sin θ ≈ θ and find where it breaks 2%.
- Given / findsin θ = θ − θ³/6 + …, with θ in radians. Find the relative error of using θ alone, and the angle where it hits 2%.
- ModelThe first dropped term is θ³/6, so the relative error of sin θ ≈ θ is about (θ³/6)/θ = θ²/6.
- Set the error to 2%θ²/6 = 0.02 gives θ² = 0.12, so θ = 0.346 rad = 19.8°.
- Check at 20°θ = 0.349 rad, sin θ = 0.342; the relative error is (0.349 − 0.342)/0.342 = 2.0%, confirming the estimate.
- Check at 5°θ = 0.0873 rad, error θ²/6 = 0.0013, about 0.13%: negligible, which is why small-angle analysis is safe for fine vibrations.
- ConclusionThe linear pendulum, and with it the natural frequency ω = √(g/L), is honest to about 2% up to roughly 20°, and far better below 10°. The Taylor series does not just permit the approximation; it tells you exactly how far it can be trusted.
Misconceptions and diagnostics
| Mistake | Symptom | Diagnostic question | Correction |
|---|---|---|---|
| Terms going to zero means it converges | The harmonic series claimed to sum | "Is shrinking terms necessary or sufficient?" | Necessary only. Some series with vanishing terms still diverge; a convergence test is needed. |
| Using a low-order series far from the point | Small-angle rule applied at 60° | "How big is the first dropped term here?" | The approximation is local. Check the leftover term over your actual range. |
| Dropping the factorials | Series terms far too large | "Did each term get divided by n!?" | The n! in the denominator is what makes the series converge; never omit it. |
| Degrees in a small-angle rule | sin θ ≈ θ wrong by a factor of 57 | "Is θ in radians?" | sin θ ≈ θ holds only in radians, where θ and sin θ share the same small-number scale. |
Practice ladder
Sum the geometric series 1 + 0.5 + 0.25 + 0.125 + … and state why it converges.
Show answer
Sum = 1/(1 − 0.5) = 2. It converges because the ratio r = 0.5 has magnitude below 1.
Then write the first three terms of the Maclaurin series of cos x.
Show answer
cos x ≈ 1 − x²/2! + x⁴/4! = 1 − x²/2 + x⁴/24.
Estimate cos(0.1) with two terms and compare to the true value 0.99500.
Show answer
1 − (0.1)²/2 = 1 − 0.005 = 0.995. The error is about the next term x⁴/24 = 4 × 10⁻⁶: essentially exact.
Use (1 + x)n ≈ 1 + nx to estimate √1.06.
Show answer
√1.06 = (1 + 0.06)1/2 ≈ 1 + 0.5(0.06) = 1.030. True value 1.0296: within 0.04%.
A vibration model uses sin θ ≈ θ. Your design swings to 30°. Estimate the relative error you are accepting, and decide whether to keep the linear model.
Show answer
θ = 0.524 rad, error ≈ θ²/6 = 0.046, about 4.6%. For a rough natural-frequency estimate this may be acceptable; for accurate amplitude prediction at 30°, keep the next term or solve the nonlinear equation numerically (Numerical Methods).
Take any formula you use with a square root, a power, or a trig function of a small quantity (relativistic mass, pendulum period, pipe-area change). Replace it with its one or two-term Taylor approximation, state the range where the error stays under 1%, and explain what the approximation reveals about the dominant behaviour.
What good work looks like
The expansion written out, the first dropped term used as the error, a stated valid range, and a sentence on the insight, such as "to first order the period depends only on length, not amplitude."
Working with AI, and proving it yourself
Use AI as an examiner, not a solver
Portfolio task
Make a one-page "Approximation Card": the four standard Maclaurin series, the three engineering shortcuts (sin θ ≈ θ, cos θ ≈ 1 − θ²/2, (1 + x)n ≈ 1 + nx), and for each shortcut the angle or x where the error reaches 1%, computed by you.
Retrieval and spaced review
Closed notes. Answer out loud, then reveal.
1. What does it mean for a series to converge, and is shrinking terms enough?
The partial sums approach a finite limit. Shrinking terms is necessary but not sufficient; the harmonic series is the counterexample.
2. Write the geometric series sum and its condition.
1 + r + r² + … = 1/(1 − r) for |r| < 1.
3. What is a Taylor series, in one sentence?
A polynomial built from a function's derivatives at a point that approximates the function near that point.
4. State the three engineering small-quantity approximations.
sin θ ≈ θ; cos θ ≈ 1 − θ²/2; (1 + x)n ≈ 1 + nx, all near zero.
5. How do you estimate the error of a truncated series?
By the size of the first dropped term, evaluated over the range of interest.
Textbook mapping
| Item | Mapping |
|---|---|
| Main source | Stewart, Calculus: Early Transcendentals (sequences, series, and Taylor series chapters) |
| Core topics | 6.1 Sequences, series, convergence · 6.2 Geometric series · 6.3 Taylor and Maclaurin series · 6.4 Engineering approximations · 6.5 Radius of convergence and error |
| Engineering connection | Dynamics and Vibrations (small-angle), Numerical Methods (how functions are evaluated), Controls, and every linearised model. |
| Skip on first pass | Convergence-test zoo, Fourier-as-series formalism, analytic continuation. Keep the geometric and Taylor series and the engineering shortcuts. |
| Read next | Multivariable Calculus. |