MATH1061 · Mathematics 1a
Series
This stream-closing calculus chapter is Taylor-centric: it is not a full ratio/root/alternating-test battery, but the ideas that MATH1061 actually examines — sequence convergence, the geometric series (the one infinite sum with a clean closed form, a/(1 − r) for |r| < 1), and above all Taylor and Maclaurin polynomials. A Taylor polynomial approximates a smooth function by matching its value and first n derivatives at a centre, so the polynomial “hugs” the curve nearby; the Lagrange remainder bounds how wrong that approximation can be. You should know the five standard Maclaurin series cold (ex, sin, cos, ln(1+x), the geometric series) and be able to build new ones by substituting, differentiating or integrating them.
What this chapter covers
- 01Sequences and their limits — where the terms settle
- 02The geometric series — converges iff |r| < 1, to a/(1 − r)
- 03Taylor & Maclaurin polynomials — match value and first n derivatives at a
- 04The standard Maclaurin series — five expansions to know cold
- 05The Lagrange remainder — bounding the approximation error
- 06The exp–log pair and convergence by the integral (p-test)
Worked example: a geometric series and a Maclaurin polynomial
- +1(a) Identify a and r: first term a = 1, common ratio r = 1/3; since |r| < 1 the series converges.
- +1(a) Apply a/(1 − r): sum = 1 / (1 − 1/3) = 1 / (2/3) = 3/2.
- +1(b) Recall the series: ex = 1 + x + x²/2! + x³/3! + ⋯, so T₃(x) = 1 + x + x²/2 + x³/6.
- +1(b) Substitute x = 0.1: 1 + 0.1 + 0.005 + 0.000167…
- +1(b) Add: ≈ 1.105167.
- +1(b) Sanity-check: the true e0.1 ≈ 1.105171, so the cubic Taylor error is about 4×10⁻⁶ — tiny because 0.1 is close to the centre 0.
Key terms
- Convergent sequence
- A sequence (an) converges to L if its terms eventually stay within any ±ε band around L. If no such L exists the sequence diverges.
- Geometric series
- A sum of terms with a constant ratio r: a + ar + ar² + ⋯. It converges if and only if |r| < 1, to the closed form a/(1 − r); otherwise it diverges. The only infinite series in the course with a clean exact sum.
- Taylor / Maclaurin polynomial
- Tn(x) about a centre a matches f's value and first n derivatives at a, so it approximates f near a. A Maclaurin polynomial is the special case centred at a = 0.
- Lagrange remainder
- The error after Tn: Rn(x) = f(n+1)(c) (x − a)n+1/(n+1)! for some unknown c between a and x. Bounding |f^(n+1)| gives a worst-case error and shows the approximation improves near the centre.
- p-test
- For an improper integral ∫₁^∞ x^(−p) dx (and the matching series), convergence holds exactly when p > 1. The clean power criterion used to bookend convergence questions in this stream.
Series FAQ
When does a geometric series converge, and what's the sum?
It converges exactly when the common ratio satisfies |r| < 1; then the infinite sum is a/(1 − r), where a is the first term. If |r| ≥ 1 the terms don't shrink to zero and the series diverges. Spotting the first term a and the ratio r correctly is the whole task — divide consecutive terms to confirm r is constant.
What does a Taylor polynomial actually approximate?
It approximates a smooth function near a chosen centre a by matching the function's value and first n derivatives there. Because the match is local, the approximation is excellent close to a and degrades as you move away — which is why estimating e^0.1 from a Maclaurin (centre 0) polynomial is very accurate, while e^5 would not be.
Do I have to derive every Maclaurin series from scratch?
No — know the five standard ones (eˣ, sin x, cos x, ln(1+x), and the geometric series 1/(1−x)) cold, then build new series by substituting, differentiating or integrating them. For example, the series for e^(x²) comes from substituting x² into the eˣ series, not from differentiating e^(x²) repeatedly.
How do I bound the error of a Taylor approximation?
Use the Lagrange remainder: the error after the degree-n polynomial is one extra term, f^(n+1)(c)(x−a)^(n+1)/(n+1)!, for some c between the centre and x. You don't know c, so bound |f^(n+1)| by its maximum on the interval and compute the worst-case size. The (x−a)^(n+1) factor is why errors shrink fast near the centre.
Exam move
Treat this as a Taylor chapter, not a convergence-test marathon. Memorise the five standard Maclaurin series and the geometric closed form a/(1 − r), and practise building new series by substitution/differentiation/integration rather than deriving from the definition. For convergence questions, identify a and r for geometric sums and apply the p-test for the integral bookend. When asked for an estimate, write the polynomial, substitute, and quote the Lagrange remainder for the error bound — the marks split between the approximation and the justified error. Keep arithmetic exact where the question allows (fractions, factorials).
