Did you know? The x² term of excos x disappears when you multiply its Maclaurin series. That does not mean there is no x³ term. The x³ coefficient survives because a first-order term in one factor can multiply a second-order term in the other. Losing that cross-term can turn a good JC2 H2 approximation into the wrong polynomial.
This Bukit Timah JC2 H2 Mathematics tutor guide answers one A-Level 9758 question: How do I multiply two Maclaurin series and keep every term through the requested power? The answer is to track total degree, not just the largest exponent visible in either factor.
Quick answer: exponents add when terms multiply
To collect the coefficient of x³ in a product A(x)B(x), consider constant × cubic, linear × quadratic, quadratic × linear and cubic × constant, wherever those terms exist. Keep contributions whose exponents sum to three. More generally, for a polynomial approximation through xm, retain all product terms with total exponent at most m.
- Expand: write the standard series for each factor to sufficient order.
- Multiply systematically: collect by powers, not by visual proximity.
- Cancel carefully: a zero x² coefficient does not prove all later coefficients are zero.
- State the order: write + O(x4) when the expansion is valid through cubic order near x = 0.
First collect the two familiar series
Near x = 0 and with angles in radians, ex = 1 + x + x²/2 + x³/6 + O(x4). The cosine series is cos x = 1 − x²/2 + O(x4); it contains no linear or cubic terms. Multiply these two expressions, retaining all contributions through x³.
The first factor does not need an x⁴ term to find the product through x³, because the second factor starts with a constant. The cosine factor does not require a cubic term because it is zero. A controlled order statement prevents students from doing many unnecessary lines of expansion.
Worked example: eˣ cos x through cubic order
Multiply (1 + x + x²/2 + x³/6)(1 − x²/2), ignoring terms of degree four and above. The constant is 1, and the linear coefficient is 1. The quadratic coefficient is (1/2) − (1/2) = 0. The cubic coefficient is (1/6) from x³/6 × 1 and −1/2 from x × (−x²/2). Their total is −1/3.
Thus the cubic-order Maclaurin approximation is excos x = 1 + x − x³/3 + O(x4). The vanished x² term and the surviving negative x³ term are the whole lesson. Forget the cross-term −x³/2, and you would incorrectly report +x³/6.
Why the x³ cross-term is easy to miss
A common shortcut is to multiply corresponding powers: constant with constant, linear with linear, quadratic with quadratic. That is not how polynomial multiplication works. The exponent of a product xj × xk is xj+k. To form x³, you must look at every pair whose exponents add to three, even if the factors are written on different lines.
A simple ledger for this example reads: x³/6 × 1 contributes +1/6; x × (−x²/2) contributes −1/2; all other x³ contributions are zero. Add them to get −1/3. This ledger is shorter and safer than distributing everything through x⁵ and then trying to delete unwanted terms.
Does the approximation give a sensible number?
At x = 0.1, the cubic polynomial gives 1 + 0.1 − 0.1³/3 = 1.0996667 to seven decimal places. The original expression e0.1cos(0.1) is approximately 1.0996497. They differ by roughly 0.0000170. That is small for a near-zero input and consistent with an omitted fourth-order term.
This agreement is a plausibility check, not a proof that the cubic approximation is equally accurate at every x. The farther x moves from the expansion point, the more neglected terms may matter. Maclaurin series describe behaviour around x = 0; a finite truncation is not automatically an exact identity.
Transfer the method: eˣ cos 2x
Now replace cos x by cos 2x. Since cos 2x = 1 − 2x² + O(x4), the product through cubic order is (1 + x + x²/2 + x³/6)(1 − 2x²). The quadratic coefficient becomes 1/2 − 2 = −3/2. The cubic coefficient becomes 1/6 − 2 = −11/6.
So excos 2x = 1 + x − (3/2)x² − (11/6)x³ + O(x4). There is no new rule: the factor of 2 inside the cosine changes the coefficient of x², which then changes the cubic cross-term with x.
One more transfer: eˣ sin x
Using sin x = x − x³/6 + O(x5), calculate (1 + x + x²/2 + x³/6)(x − x³/6). Keep terms through x³. The result is exsin x = x + x² + x³/3 + O(x4). The x³ coefficient comes from x²/2 × x and 1 × (−x³/6): 1/2 − 1/6 = 1/3.
The result begins at order x because sin 0 = 0. Checking the constant term from the actual function is a quick audit: e⁰sin 0 = 0, so any expansion with constant term 1 is wrong before you examine the rest.
Where the approximation is valid
The exponential, sine and cosine Maclaurin series converge for all real x, but a truncated polynomial is usually most accurate near x = 0. Other standard series have restrictions, such as the binomial and logarithmic expansions. Never transfer the unrestricted convergence of ex to an unrelated series.
For example, ln(1 + x) = x − x²/2 + x³/3 + ··· is valid in the usual Maclaurin expansion region |x| < 1. If the expression also contains 1/(1 − x), inspect both series’ domains before multiplying them. The 2027 H2 syllabus explicitly includes the convergence range of standard series.
The five mistakes to remove before prelims
- Forgetting cross-terms: assemble each output coefficient from every exponent pair that adds to that degree.
- Dropping a nonzero higher-order term after cancellation: x² may vanish while x³ survives.
- Using degrees instead of radians: the standard sine and cosine Maclaurin coefficients require radian arguments.
- Confusing polynomial equality with approximation: show the remainder or use ≈ and specify near x = 0.
- Forgetting a domain condition: confirm where the standard series used actually converges.
One-minute Maclaurin diagnostic
Write two rows: ex has x³ coefficient +1/6; cos x has x² coefficient −1/2 and zero x³ coefficient. Ask: “Where else can an x³ term come from?” A student should point to x × (−x²/2) immediately. Then change cos x to cos 2x and ask which coefficient changes. If the answer is the x² coefficient, the learner is beginning to track structure instead of memorising one final expansion.
Eight quick checks with answers
- 1. The x² coefficient of excos x is 0.
- 2. The x³ coefficient of excos x is −1/3.
- 3. The x² coefficient of cos 2x is −2.
- 4. The x³ coefficient of excos 2x is −11/6.
- 5. The first nonzero power of exsin x is x.
- 6. Its x³ coefficient is 1/3.
- 7. When multiplying x² by x³, the result has degree 5.
- 8. Is 1 + x − x³/3 an exact formula for excos x? No; it is an approximation near x = 0.
How a good JC2 tutorial makes the series manageable
A useful teaching sequence is to start with a row for each factor, collect x⁰ through x³ in separate columns, then ask one learner to audit all pairs contributing to the cubic column. Another learner estimates the function at x = 0.1, while a third explains why the approximation worsens far away from zero. The method develops calculation, reasoning and validation together.
For the wider chapter, read the Bukit Timah H2 Maclaurin series and small-angle approximations guide. This article specifically teaches the product cross-term that frequently disappears in rushed algebra.
Frequently asked questions
Why is there no x² term in eˣ cos x?
Because +x²/2 from ex cancels −x²/2 from cos x in the product.
Does a missing x² term imply the cubic term is also missing?
No. The cubic coefficient receives contributions from other combinations of powers, and here it equals −1/3.
How many terms should I write in each series?
Enough to capture every term up to the requested total degree. You do not need an arbitrary number of extra powers when their products cannot contribute.
Why use O(x⁴) after a cubic-order expansion?
It signals that the omitted remainder is bounded in magnitude by a constant multiple of |x|⁴ near zero, for the analytic functions used in this example.
Is multiplying standard series part of the 2027 H2 syllabus?
Yes. Maclaurin series, using standard series and deriving first terms of products such as an exponential times a trigonometric function are within syllabus 9758.
Your next JC2 H2 Mathematics question
- Area Between Crossing Curves: Why You Must Split
- Maclaurin Products: The Cross-Term Trap — this guide
- Stationary Inflexion or Turning Point?
- Differential Equations: Don’t Lose Equilibrium Solutions
For a complete subject overview, use the Bukit Timah H2 Mathematics tuition guide and the Bukit Timah JC2 tuition pathway. The eduKateSG small-group mathematics tutoring reference explains the three-student teaching format used in a different location-specific example. The worksheets and figures in this article are fictional and illustrative, not real pupil results.
Official source and scope
Reference: Singapore Examinations and Assessment Board, H2 Mathematics syllabus 9758 (2027), official PDF. This page teaches a precise syllabus skill, rather than claiming to replace the full syllabus. All worked problems and quick checks here were written as instructional examples; they are not past-year examination questions or verified claims of student achievement.