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Maclaurin series and approximation

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By the end, you can

  • Build expansions from derivatives at zero
  • Substitute and combine standard series
  • State convergence and the order of an approximation

about an hour for the lesson · practise at your own pace.

The central idea

A Maclaurin expansion matches a function near zero with f(0)+f′(0)x+f′′(0)x2/2!+⋯f(0)+f\prime(0)x+f\prime\prime(0)x^2/2!+\cdots. Standard examples include exe^x, sine, cosine, ln⁡(1+x)\ln(1+x) and the binomial expansion. A polynomial truncation is an approximation; the infinite expansion also has a convergence domain.

Why coefficients contain factorials

Suppose f has a power-series representation near zero that can be differentiated term by term. From f(x)=a0+a1x+a2x2+⋯f(x)=a_0+a_1x+a_2x^2+\cdots, substitution at zero after successive derivatives gives f(0)=a0f(0)=a_0, f′(0)=a1f^{\prime}(0)=a_1, f′′(0)=2!a2f^{\prime\prime}(0)=2!a_2 and f′′′(0)=3!a3f^{\prime\prime\prime}(0)=3!a_3. In general f(n)(0)=n!anf^{(n)}(0)=n!a_n, so an=f(n)(0)/n!a_n=f^{(n)}(0)/n!.

f(x)=f(0)+xf′(0)+x22!f′′(0)+⋯+xnn!f(n)(0)+⋯ .f(x)=f(0)+xf^{\prime}(0)+\frac{x^2}{2!}f^{\prime\prime}(0)+\cdots+\frac{x^n}{n!}f^{(n)}(0)+\cdots.

This is the Maclaurin series listed in MF27. The derivation assumes a valid power-series representation; having derivatives of all orders alone does not guarantee equality to the series. State an applicable interval.

Calculate a function using elementary arithmetic

The source asks how sin⁡(0.2)\sin(0.2) could be calculated without a computer or calculator. A suitable polynomial approximation reduces evaluation to addition, subtraction, multiplication and division. Its displayed value is sin⁡(0.2)=0.19866933…\sin(0.2)=0.19866933\ldots (radians). The source uses this calculation to motivate replacing a difficult function evaluation with elementary arithmetic.

The chapter outline names standard expansions of (1+x)n(1+x)^n for rational n, exe^x, sine, cosine and ln⁡(1+x)\ln(1+x); derivation by repeated differentiation (such as sec⁡x\sec x), repeated implicit differentiation (such as y3+y2+y=x2−2xy^3+y^2+y=x^2-2x), and combining standard series (such as excos⁡2xe^x\cos2x and ln⁡((1+x)/(1−x))\ln((1+x)/(1-x))). It also names convergence intervals, approximation and the small-angle forms sin⁡x≈x\sin x\approx x, cos⁡x≈1−x2/2\cos x\approx1-x^2/2, tan⁡x≈x\tan x\approx x.

The Student copy is incorporated through printed page 92 and the Teacher through printed page 91, including the standard derivations, Examples 1–11 and Self-Practice 1–4. Further techniques remain pending.

Compare the logarithm with successive partial sums

Build the logarithm one power at a time

Compare ln(1 + x) with one partial sum at a time. Choose a source degree, then move x and inspect the numerical error.

Series and source notes

P₀ = 0; P₁ = x; P₂ = x − x²/2; P₃ = x − x²/2 + x³/3; P₄ = x − x²/2 + x³/3 − x⁴/4.

The infinite series equals ln(1 + x) for −1 < x ≤ 1. At x = 1, the alternating harmonic series converges to ln 2. At x = −1, the function is undefined and the series diverges. For x > 1, the terms do not tend to zero. The shading marks only the open interval; its endpoints need separate checks.

The original sheet adds several polynomials before isolating P₄. This explorer compares one polynomial with the function. Original selections and colours remain in the linked source diagram. The initial window is illustrative, not an original calculator setting; the vertical range expands to include the selected values. Visual agreement is not a convergence proof.

Five mathematical reconstructions retain the source selections: logarithm and linear polynomial; add the quadratic, then cubic, then quartic partial sum; finally show only the logarithm and quartic. Colours match the printed screens. Illustrative common window −0.95≤x≤1.4,−2.5≤y≤1.8; visual closeness is not a proof of convergence.
Open original diagram (new tab)

Compare cosine with successive even-power partial sums

Build cosine with even powers

Use radians. Compare the constant, quadratic, quartic and sixth-degree partial sums with cosine, then move away from zero to inspect the error.

Series and source notes

P₀ = 1; P₂ = 1 − x²/2!; P₄ = P₂ + x⁴/4!; P₆ = P₄ − x⁶/6!. Odd-power coefficients vanish, so P₁ = P₀, P₃ = P₂, and P₅ = P₄.

The cosine series converges for every real x in radians. Shading shows a local comparison window, not a convergence interval or accuracy guarantee. A fixed truncation can have a large error far from zero.

The original sheet adds several partial sums before isolating P₆. This explorer compares one chosen truncation with cosine. Original selections and colours remain in the linked source diagram. The initial window is illustrative; the vertical range expands to include the selected values. Visual agreement near zero is not a convergence proof.

Five reconstructed stages compare blue cosine with the constant 1, then quadratic, quartic and sixth-degree partial sums; the last stage hides earlier polynomials. Source colours and selections are retained, including the constant changing from red to magenta. Illustrative common window minus 7 to 7 horizontally and minus 3 to 3 vertically; local agreement does not prove convergence.
Open original diagram (new tab)

Explore

For e2xe^{2x}, compare the quadratic polynomial P2=1+2x+2x2P_2=1+2x+2x^2 with P4=P2+4x3/3+2x4/3P_4=P_2+4x^3/3+2x^4/3. Predict which is closer at x=0.1x=0.1, then compare both errors using a calculator or graph.

Repeat at x=0.5x=0.5 and x=2x=2 using the same scale for all three curves at each input.

Explain why convergence of the exponential series for all real x does not mean a fixed low-degree polynomial is accurate at every input. The numerical comparisons assess these inputs, without proving a general error bound.

Related interactive example

Explore this related example, then return to the prediction task above.

Adding terms to 2(1 − 3x/4)^(1/2)

Compare the function with a finite polynomial. Change the highest included power and test x, then use the error plot to see how the approximation changes at that x.

Highest included powerxⁿ, n = 2
Test value xx = 0.25
Reset the controls
Function
1.80278
P₂(x)
1.80371
Error
0.00094

Inside the stated open interval. A finite truncation can still have a noticeable error.

Teal marks the function, amber marks the polynomial, and the red segment joins their values at the selected x.

Does keeping more powers help at this x?

Highest included power n runs horizontally; signed error Pₙ(x) − f(x) runs vertically. Zero means agreement at this x. A missing power leaves the polynomial unchanged; individual steps need not improve the error.

  • Function
  • Series truncated at the chosen power
  • Stated open interval. Endpoint convergence requires a separate check
Key idea

Inside −43<x<43-\frac43<x<\frac43 extra terms improve the approximation; the displayed truncated polynomial approximates the function there.

Ideas to try
  1. Slide xx towards 43\frac43: does the polynomial still approach the curve?
  2. Move xx beyond 43\frac43 and see what happens to the readouts.
Open the full interactive problem →

Worked example

e2x=1+2x+2x2+43x3+⋯e^{2x}=1+2x+2x^2+\frac43x^3+\cdots. The quadratic approximation at x=0.1 is 1+0.2+0.02=1.221+0.2+0.02=1.22. The omitted terms begin at order x³.

An expansion valid near zero does not automatically justify approximation at a large input.

Check your understanding

For small nonzero x, obtain the expansion of (e3x−1−3x)/x2(e^{3x}-1-3x)/x^2 through the term in x2x^2. Explain how many terms of the numerator you need, use the result to estimate the quotient at x=0.1x=0.1, and distinguish the original domain from the limit at zero.

Mastery learning objectives

At the end of this chapter, I should be able to:

Derive and combine series

Derive a Maclaurin series by repeated differentiation, for example sec⁡x\sec x.

Derive a Maclaurin series by repeated implicit differentiation, for example y3+y2+y=x2−2xy^3+y^2+y=x^2-2x.

Combine standard series, for example excos⁡(2x)e^x\cos(2x) and ln⁡((1+x)/(1−x))\ln((1+x)/(1-x)).

Convergence and validity

Determine the range in which a standard series converges.

Determine the range in which a derived expansion is valid.

Check and apply

Check a derived Maclaurin series using standard series.

Use a suitable substitution to approximate an irrational number.

Use small-angle approximations to approximate expressions.

Choose a practice question

The six rows retain the printed study map on Student page94 and Teacher page93. Follow a link to the whole question, then attempt the named parts before opening the working.

SkillLecture examplesDiscussionReview follow-up
Binomial expansion: perform expansion3, 4, 5(i), 8(iv)2, 3, 4, 5, 6(i)1, 2, 3
Binomial expansion: find validity3, 4, 5(ii)3, 41, 2
Binomial expansion: approximate a value5(iii)(iv)(v)6(ii)(iii)3(i)(ii)
Maclaurin: use standard MF27 results1, 2, 8(iii)5, 7(iii)5, 7(ii), 10(ii)
Maclaurin: derive terms6, 7, 8(i)(ii)7(i)(ii), 8, 9, 10, 11(i)(ii)4, 5, 6, 7(i), 8, 9, 11
Small-angle approximations9, 1011(iii), 12—

The source labels the approximation row “Determine approximate value of a rationale function”; the linked tasks estimate irrational numbers, so use the actual asks. Discussion11(iii) is retained in the printed small-angle row, but asks for coefficient matching in e2xcos⁡xe^{\sqrt2x}\cos x. Discussion6(iii) retains its qualified target-substitution ambiguity. A dash represents the source’s blank Review cell.

Practice questions

Practice section 1 · Questions 1–6

Expand a logarithm after changing its argument · Transform the interval along with the series argument · +4 more

Practice section 2 · Questions 7–12

Build a fourth-degree series from a differential equation · Differentiate a tangent expansion to obtain a secant-square series · +4 more

Practice section 3 · Questions 13–18

Track the order of an approximation through division · Normalize a square root before using the binomial series · +4 more

Practice section 4 · Questions 19–24

Factor out a negative power before expanding · Match a product-series coefficient · +4 more

Practice section 5 · Questions 25–30

Transform an exponential series without differentiating again · Match trigonometric and exponential series · +4 more

Practice section 6 · Questions 31–36

Use a derivative identity for an even exponential composition · Verify an exponential–sine series by two methods · +4 more

Practice section 7 · Questions 37–39

Compare a cubic approximation with the actual function · Use the printed half-unit initial value in a derivative recurrence · +1 more

More practice

State the substituted convergence conditions and requested final order before expanding. Separate an exact identity, a convergent series representation, a finite approximation and any claim about accuracy.

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