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Mathematics · University

Taylor series, convergence and error

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

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Prerequisites

Limits, derivatives and infinite series.

Study notes

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

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

The coefficient is 2³/3!=8/6=4/3.

4/3

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

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

    1. It diverges
    2. The ratio test proves endpoint convergence
    3. It equals zero
    4. It must converge because the radius is 1

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

    1. It converges absolutely
    2. It has no terms
    3. It diverges because all negative terms diverge
    4. It converges conditionally

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

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

Answers and marking guidance

Answers and marking notes
  1. It diverges

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

  2. It converges conditionally

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

  3. 1.105

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

  4. 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.

Further reading and review references

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