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Mathematics · lesson 2 of 4

Taylor series, convergence and error

The coefficient of (x−a)^n is f⁽ⁿ⁾(a)/n!, not just the derivative.

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Learn the ideas

Prerequisites: Limits, derivatives and infinite series.

Coefficients

The coefficient of (x−a)^n is f⁽ⁿ⁾(a)/n!, not just the derivative.

Radius

The radius of convergence controls an open interval or disk; endpoints require separate tests.

Remainder

A bound on the next derivative over the relevant interval yields a Taylor remainder bound.

Alternation

An alternating-series error bound requires terms decreasing in magnitude to zero.

Worked examples

Example 1

Find the coefficient of x³ in the Maclaurin series of exp(2x).

The coefficient is 2³/3!=8/6=4/3.

4/3

Example 2

Find the radius of convergence of the power series sum from n=0 to infinity of (x/4)^n.

This geometric series converges for abs(x/4)<1.

4

Check your understanding

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

1 Find the coefficient of x³ in the Maclaurin series of exp(2x).

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4/3

The coefficient is 2³/3!=8/6=4/3.

2 Find the radius of convergence of the power series sum from n=0 to infinity of (x/4)^n.

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4

This geometric series converges for abs(x/4)<1.

3 The power series sum x^n/n, n≥1, has radius 1. What happens at x=1?

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It diverges

At x=1 this is the harmonic series. The radius alone does not settle endpoints.

4 Does the series sum from n=1 to infinity of (−1)^n/n converge absolutely, conditionally, or not at all?

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It converges conditionally

The alternating harmonic series converges but its absolute-value series diverges.

5 Approximate exp(0.1) by 1+x+x²/2. Give the polynomial value.

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1.105

Substitution gives 1+0.1+0.005=1.105; this is an approximation to exp(0.1).

6 Why does differentiating an infinite series term by term require a theorem rather than just linearity for finite sums?

Sample answer and marking guidance

An infinite series involves a limiting operation. Interchanging that limit with differentiation needs hypotheses, such as those valid inside a power series radius.

Award credit for distinguishing finite sums from limits and mentioning applicable convergence or power-series hypotheses.

7 For the alternating series 1−1/2+1/3−1/4+..., what upper bound on absolute error follows after the fourth term?

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0.2

The positive term magnitudes decrease to zero, so the next magnitude 1/5 bounds the remainder.

8 Use a geometric-series identity to derive 1/(1−x)=sum x^n for abs(x)<1.

Sample answer and marking guidance

The partial sum is (1−x^(N+1))/(1−x). Since x^(N+1) tends to zero, the limit is 1/(1−x).

Credit the finite-sum identity, the domain abs(x)<1, and the vanishing remainder.

Further reading and review references

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