Taylor Series & Asymptotic Thinking (IPhO Math Tools)

In IPhO, Taylor series are rarely about summing infinitely many terms. They are about seeing a small parameter, expanding only as far as needed, and turning a messy expression into a usable leading-order result.

1. Definitions (Must Know)

A. Taylor and Maclaurin expansions

Taylor expansion of f(x) about x₀:

Maclaurin expansion is the special case x₀ = 0.

In timed work, you usually keep only a few terms:

B. Big-O and “order kept”

Writing

means “everything omitted is at most of order x³ near the expansion point”.

C. Asymptotic equivalence

As x → ∞, the statement

means f(x)/g(x) → 1.

This is about dominant behaviour, not necessarily convergence of a power series.

D. Small parameter thinking

In physics, your expansion variable is often a dimensionless ratio:

• β = v/c (relativistic corrections)
• ε = x/L (small displacement compared with system size)
• δ = (Δ T)/T₀ (small fractional change)

2. Key Ideas (What Earns Marks)

• Identify the dimensionless small parameter first, then expand.
• Keep orders consistent: if you keep O(ε²) terms in one place, do not drop them elsewhere in the same derivation.
• Use symmetry to predict which terms vanish (odd or even behaviour).
• For large-parameter asymptotics, rewrite in terms of a small variable like 1/x.
• Always state the validity: “for |x|≪ 1” or “for x≫ 1”.
• Check your approximation by taking a limit (does it reduce to a known simple case?).

3. Detailed Explanations

A. A minimal expansion toolkit (high yield)

For |x|≪ 1:

• Exponential:
  e^x = 1 + x + x²/2 + x³/6 + O(x⁴)
• Sine and cosine:
  sin x = x-x³/6 + O(x⁵), cos x = 1-x²/2 + x⁴/24 + O(x⁶)
• Logarithm:
  ln(1 + x) = x-x²/2 + x³/3 + O(x⁴) (|x| < 1)
• Binomial:
  (1 + x)^α = 1 + α x + ((α(α-1))/2)x² + O(x³)
• Reciprocal and square root:
  1/(1-x) = 1 + x + x² + O(x³) (|x| < 1)
  square root of (1 + x) = 1 + x/2-x²/8 + O(x³)

Memorizing these saves real time.

B. Linearization (the physics meaning of first order)

If x = x₀ + δ x with |δ x|≪ 1 (in the right units), then

This is why “small oscillations” lead to linear equations: you Taylor expand a force about equilibrium and keep the first nonzero term.

C. Large-variable asymptotics via a small ratio

If x is large compared with a, you should rewrite:

Now you expand in (a/x)².

D. Exponentials from limits (discrete to continuous)

The classic approximation

is essentially a Taylor expansion of ln(1 + u) with u = x/n.

This shows up whenever many small multiplicative changes accumulate (collisions, repeated attenuation steps, compound processes).

E. When Taylor is not a convergent “power series method”

Some asymptotic expansions are not convergent, but the first few terms still give excellent approximations in the relevant limit. In olympiad settings, you typically only need the leading one or two terms, plus a clear statement of the regime.

4. Common Mistakes

• Expanding in a dimensional variable instead of a dimensionless ratio.
• Using sin x ≈ x with x in degrees.
• Keeping a second-order term in one expression but dropping an equally sized term elsewhere.
• Expanding around the wrong point (for example around 0 when the natural point is x₀).
• Forgetting to state the validity range, then using the approximation outside it.

5. Exam Tips

• Write “Let ε be small” and define it. Examiners like seeing the expansion parameter.
• Mark the order you keep, for example “keeping up to O(ε²)”.
• If you get a correction term, check its sign in a physical limit (does the correction increase or decrease the quantity as expected?).
• If you need only a leading-order estimate, stop early and move on.

6. Worked Examples

A. Relativistic kinetic energy for v≪ c

Starting from γ = 1/(square root of (1-β²)) with β = v/c, expand the kinetic energy K = (γ-1)mc² for small β.

B. Solving cos θ = 1-ε for small ε

If ε is small and positive, estimate θ.

C. A large-x simplification that saves algebra

For x≫ a > 0, approximate square root of (x² + a²) -x.

7. Mind Stretchers

A. Small-angle correction to the pendulum period

The exact period of a simple pendulum of maximum angle θ₀ can be written

Show that for small θ₀, the period increases as

to leading correction.

8. Practice