Sequences, Infinite Series, and Taylor Models

Introduction and context

Engineering models often replace a physical process with a sequence of numerical approximations. A controller may update an estimate at discrete sampling instants; a numerical solver may generate successive iterates; and a nonlinear function such as sin θ, eˣ, or ln(1 + x) may be replaced by a polynomial that is easier to evaluate in real time. The mathematical language for these ideas is the theory of sequences, infinite series, and Taylor models.

A sequence is an ordered list of numbers indexed by an integer. A series is the sum of the terms of a sequence. Convergence determines whether an approximation approaches a finite, meaningful value. Taylor and Maclaurin series provide systematic polynomial representations of differentiable functions, while remainder estimates quantify the error introduced by truncation.

In mechatronics, these ideas support numerical algorithms, sensor calibration, motion planning, system identification, embedded computation, and local linearization of nonlinear models.

Learning objectives

By the end of this chapter, you should be able to:

• distinguish sequences from series and determine sequence limits;
• apply the term, comparison, limit-comparison, ratio, root, integral, and alternating-series tests;
• distinguish absolute from conditional convergence;
• determine the radius and interval of convergence of a power series;
• construct Taylor and Maclaurin polynomials and series;
• estimate truncation error using Taylor's theorem;
• select a suitable polynomial approximation for an engineering calculation;
• recognize the limitations of local polynomial models.

1. Sequences and their limits

A sequence is a function whose domain is usually the positive integers. It is written

where aₙ is the nth term. For example, a_n = 1/n gives the sequence 1, 1/2, 1/3, …. A sequence may be defined explicitly, recursively, or by an engineering algorithm.

A sequence {a_n} converges to L if its terms become arbitrarily close to L as n increases. Formally, for every ε > 0, there exists an integer N such that

Here ε is a prescribed tolerance, and N may depend on ε. We write lim_n→∞ a_n = L. If no finite L exists, the sequence diverges.

1.1 Common methods for finding limits

For rational functions of n, divide numerator and denominator by the highest power of n. For example,

A sequence of the form rⁿ converges to 0 when |r| < 1, diverges when |r| > 1, and is constant or oscillatory in special cases. Thus (−1)ⁿ does not converge because it alternates between −1 and 1.

The squeeze theorem is useful when a sequence is bounded between two sequences with the same limit. If b_n ≤ a_n ≤ c_n and both b_n and c_n tend to L, then a_n also tends to L.

A sequence is increasing if a_n+1 ≥ a_n and decreasing if a_n+1 ≤ a_n. It is bounded above if there is a number M such that a_n ≤ M for all n, and bounded below if a_n ≥ m. The monotone convergence theorem states that every increasing sequence bounded above, or decreasing sequence bounded below, converges.

Worked example 1: a recursive sequence

Consider a₁ = 1 and

The first terms are 1, 1.5, 1.75, and 1.875. The sequence is increasing and bounded above by 2: if a_n < 2, then a_n+1 < 2. Therefore it converges. Let its limit be L. Taking limits on both sides gives

so 2L = L + 2 and L = 2. This iteration is an example of a numerical process converging to a fixed point.

2. Infinite series

Given a sequence {a_n}, the associated infinite series is

Its meaning is defined through partial sums:

The series converges to S if lim_N→∞ S_N = S. Otherwise it diverges. The terms of a convergent series must satisfy the necessary condition

This condition is not sufficient: the harmonic series ∑_n=1^∞ 1/n diverges even though its terms approach zero.

2.1 Geometric series

For a first term a and common ratio r,

The condition |r| < 1 is essential. If |r| ≥ 1, the series does not converge. The finite partial sum, for r ≠ 1, is

The remainder after N + 1 terms is R_N = ar^N+1/(1 − r), provided |r| < 1.

2.2 p-series

The p-series is

It converges if and only if p > 1. It diverges for p ≤ 1. This result is a principal benchmark for comparison tests.

3. Convergence tests

No single test is best for every series. The form of the term should guide the choice.

3.1 Comparison test

Suppose 0 ≤ aₙ ≤ bₙ for sufficiently large n. If ∑ b_n converges, then ∑ a_n converges. If 0 ≤ bₙ ≤ aₙ and ∑ b_n diverges, then ∑ a_n diverges.

3.2 Limit-comparison test

For positive terms, if

where 0 < c < ∞, then ∑ a_n and ∑ b_n either both converge or both diverge. Rational expressions in n are often compared with a p-series by retaining their highest powers.

3.3 Integral test

If f(x) is continuous, positive, and decreasing for x ≥ 1, and a_n = f(n), then ∑ a_n and ∫₁^∞ f(x) dx have the same convergence behavior. If the series converges, the remainder after N terms satisfies

3.4 Ratio test

For a series with nonzero terms, define

If L < 1, the series converges absolutely. If L > 1, or if the limit is infinite, it diverges. If L = 1, the test is inconclusive. Factorials and powers commonly indicate the ratio test.

3.5 Root test

Define

The same conclusions apply: L < 1 gives absolute convergence, L > 1 gives divergence, and L = 1 is inconclusive. The root test is especially effective when the nth term contains an nth power.

3.6 Alternating-series test

A series of the form

converges if bₙ ≥ 0, bₙ is eventually decreasing, and lim_n→∞ b_n = 0. Its remainder after N terms satisfies

This is convergence, but not necessarily absolute convergence. For example, the alternating harmonic series converges, whereas the harmonic series of absolute values diverges.

3.7 Absolute and conditional convergence

A series ∑ a_n converges absolutely if ∑ |a_n| converges. Absolute convergence implies ordinary convergence. A series that converges but whose absolute-value series diverges is conditionally convergent.

Rearranging terms is safe for absolutely convergent series. For conditionally convergent series, rearrangement can change the sum or destroy convergence. This matters in numerical computation because careless reordering and rounding can produce different results.

Worked example 2: selecting a test

Determine the behavior of

Compare it with 1/n². The limit comparison is

Since the limit is positive and finite, the given series has the same behavior as ∑ 1/n², which converges. Therefore the given series converges.

4. Power series

A power series centered at c has the form

The coefficients aₙ are constants and x is the independent variable. There is a radius R, possibly 0 or ∞, such that the series converges absolutely when |x − c| < R, diverges when |x − c| > R, and must be tested separately at the endpoints x = c ± R.

For a power series, the ratio test commonly gives

when this limit exists. The interval of convergence is obtained only after checking both endpoints.

Worked example 3: interval of convergence

Find the interval of convergence of

Using the ratio test on absolute values,

Thus convergence occurs when |x − 2| < 1, giving 1 < x < 3. At x = 1, the series becomes ∑(−1)ⁿ/n, which converges by the alternating-series test. At x = 3, it becomes ∑1/n, which diverges. Therefore the interval is [1,3).

Within its interval of convergence, a power series may be differentiated or integrated term by term. The resulting series has the same radius of convergence, although endpoint behavior may change.

5. Taylor and Maclaurin series

If a function f has derivatives of all required orders near x = c, its Taylor polynomial of degree n about c is

Here f⁽ᵏ⁾(c) is the kth derivative evaluated at c, k! is the factorial of k, and (x − c) is the displacement from the expansion point. The corresponding Taylor series is

The Maclaurin series is the special case c = 0:

provided the series converges to f(x) on the stated domain.

Important standard series include:

Angles in trigonometric series must be measured in radians. If θ is supplied in degrees, convert using θ_rad = θ_degπ/180.

5.1 Constructing a series from a known series

Known series can be differentiated, integrated, shifted, or multiplied by constants within their valid domains. For example, integrating the geometric series gives

so

Substitution then produces series for related functions, but the resulting convergence domain must still be checked.

6. Error bounds and polynomial approximation

A Taylor polynomial is finite and therefore computationally useful, but it introduces truncation error

Taylor's theorem in Lagrange form states that if f has n + 1 continuous derivatives between c and x, then for some ξ between c and x,

If |f⁽ⁿ⁺¹⁾(t)| ≤ M throughout the interval between c and x, then

This bound is guaranteed but may be conservative. It is valid only when the derivative bound M genuinely applies on the entire interval.

For an alternating series whose term magnitudes decrease to zero, the first omitted term gives a sharper practical bound:

Worked example 4: approximating sin x

Approximate sin(0.2) using the cubic Maclaurin polynomial and bound the error. Since x is in radians,

At x = 0.2,

The next term has magnitude

Because the sine series is alternating with decreasing term magnitudes at x = 0.2, the absolute error is at most approximately 2.667 × 10⁻⁶. The approximation is therefore suitable when an error of a few parts per million is acceptable.

Worked example 5: exponential approximation with a derivative bound

Approximate e⁰·¹ by the degree-three Maclaurin polynomial:

Thus

For eˣ, every derivative is eˣ. On [0, 0.1], M = e⁰·¹ < 1.106. Therefore

The degree-three model has a guaranteed absolute error below approximately 0.00000461 on this point.

7. Taylor models in mechatronics

A Taylor model is not merely a symbolic expansion; it is an approximation used inside a larger engineering decision. Three questions should always be asked:

1. Around which operating point is the model constructed?
2. Over what input range is the error acceptable?
3. What numerical precision and computational cost are available?

For a multivariable function f(x, y), the first-order model about (x₀, y₀) is

The partial derivatives represent local sensitivities. In vector notation, for a differentiable function f: ℝᵐ → ℝᵖ,

where J_f is the p × m Jacobian matrix. This linearization is foundational in robot kinematics, state estimation, sensor calibration, and nonlinear control.

A second-order scalar model includes curvature:

The linear model is efficient but can be inaccurate far from the operating point. A polynomial of higher degree can reduce truncation error, but it may increase computation and can behave poorly outside the expansion interval.

8. Applied case: real-time actuator compensation

Background

Consider an illustrative servo actuator whose angular sensor reports voltage V related to angle θ by the nonlinear calibration model

where θ is in radians and V is in volts. A microcontroller must evaluate the model repeatedly at small angles near θ = 0, but floating-point trigonometric evaluation is relatively expensive on the selected embedded processor.

Problem

Determine a polynomial model suitable for |θ| ≤ 0.2 rad with an absolute voltage error below 7 × 10⁻⁶ V due solely to truncating the sine series.

Analysis and approach

Use

If the cubic approximation is used, the omitted sine term is bounded by θ⁵/5! because the series is alternating with decreasing terms on this interval. At the largest magnitude θ = 0.2,

Multiplication by 2.5 gives a voltage error bound

which is below 7 × 10⁻⁶ V. Therefore the implemented model can be

Outcome

The cubic model meets the stated truncation-error requirement over the specified operating range. In a real design, the engineer must also include sensor noise, coefficient quantization, round-off, temperature drift, and calibration uncertainty. The mathematical truncation bound alone does not guarantee total system accuracy.

Learning points

• The operating interval is as important as the polynomial degree.
• Units and angle conventions must be explicit.
• A guaranteed bound should be compared with the complete error budget.
• Validation with test data remains necessary after analytical approximation.

9. Numerical and practical limitations

Cancellation and rounding

Subtracting nearly equal numbers can cause loss of significant digits. Alternative series forms or range transformations may improve numerical stability. For example, evaluating a function through a mathematically equivalent but numerically stable expression can be more important than adding another Taylor term.

Local validity

A Taylor polynomial is centered at a point. Its accuracy generally decreases as |x − c| grows. For a wide operating range, use piecewise approximations, select a better center, or apply range reduction.

Convergence is not the same as usefulness

A series may converge very slowly. The harmonic-like alternating series converges, but many terms may be required for high accuracy. Conversely, a rapidly convergent series may be computationally efficient. Engineering selection requires considering error, memory, execution time, and robustness.

Floating-point implementation

In embedded systems, evaluate polynomials using Horner's method. For

Horner's form is

It reduces the number of multiplications and often improves numerical behavior. Coefficients should be stored with adequate precision, and the approximation should be tested at interval endpoints and representative interior points.

10. Best-practice workflow

1. Identify the sequence, series, or function and its domain.
2. Check the necessary term condition before applying a convergence test.
3. Choose a test based on structure: comparison for algebraic decay, ratio for factorials, root for nth powers, and alternating test for sign changes.
4. For power series, determine the radius first and test endpoints separately.
5. State whether convergence is absolute or conditional.
6. For Taylor models, specify the center, degree, input units, and valid interval.
7. Derive or bound the remainder on the complete operating interval.
8. Include numerical, measurement, and modeling errors in the engineering error budget.
9. Validate the approximation independently, preferably with interval tests and higher-precision reference calculations.

Chapter Summary

Sequences describe ordered numerical processes, and their convergence concerns the limit of individual terms. Infinite series are defined through partial sums; a convergent series must have terms tending to zero, but that condition alone is insufficient. Geometric series and p-series provide fundamental benchmarks. Comparison, limit comparison, integral, ratio, root, and alternating-series tests offer complementary methods for determining convergence.

A power series converges inside a radius, diverges outside it, and requires separate endpoint analysis. Taylor polynomials approximate a function using derivatives at an expansion point; Maclaurin polynomials are centered at zero. Taylor's theorem supplies a remainder expression and a practical error bound. Alternating-series estimates can be especially effective for trigonometric approximations.

In mechatronics, Taylor models support local linearization, calibration, control, and embedded implementation. Their usefulness depends on operating range, computational resources, numerical stability, and the complete error budget. Convergence analysis and error estimation therefore connect mathematical theory directly to reliable engineering design.

Key Terms

Sequence: An ordered list of numbers indexed by integers.

Limit: The value approached by a sequence or function.

Partial sum: The sum of the first finite number of terms of a series.

Infinite series: The limiting process formed by successive partial sums.

Convergence: Approach to a finite limit.

Divergence: Failure to approach a finite limit.

Geometric series: A series whose consecutive terms have a constant ratio.

p-series: The series ∑1/n^p, convergent exactly when p > 1.

Comparison test: A test that uses inequalities with a known positive series.

Ratio test: A test based on the limit of consecutive-term magnitudes.

Root test: A test based on nth roots of term magnitudes.

Absolute convergence: Convergence of the series of absolute values.

Conditional convergence: Convergence without absolute convergence.

Power series: A series of powers of displacement from a center.

Radius of convergence: The distance from the center within which a power series converges absolutely.

Taylor polynomial: A finite derivative-based polynomial approximation.

Maclaurin series: A Taylor series centered at zero.

Remainder: The difference between a function and its finite approximation.

Jacobian: The matrix of first partial derivatives of a vector-valued function.

Horner's method: A nested procedure for efficient polynomial evaluation.

Review Questions

Short Answer Questions

1. Define convergence of a sequence using ε and N.
2. State the necessary term condition for convergence of an infinite series.
3. When does a p-series converge?
4. What is the difference between absolute and conditional convergence?
5. Why must the endpoints of a power-series interval be tested separately?

Long Answer Questions

1. Explain the ratio, root, comparison, and alternating-series tests, including the conditions under which each is inconclusive.
2. Derive the Maclaurin series for ln(1+x) by integrating a suitable known series and state its convergence domain.
3. Explain Taylor's theorem in Lagrange form and discuss how it produces an engineering error bound.
4. Compare local linearization with a second-order Taylor model in a nonlinear mechatronics application.
5. Discuss why convergence alone does not guarantee that a series is computationally efficient or numerically stable.

Analytical / Application Questions

1. Determine the convergence behavior of ∑_n=1^∞ (n²+1)/(n⁴+3), identifying an appropriate comparison series.
2. Find the degree-three Maclaurin approximation for e^x at x = 0.2 and use Taylor's theorem to obtain a numerical error bound on [0, 0.2].
3. A sensor model is y = ln(1+x) for |x| ≤ 0.1. Select a polynomial degree that guarantees an absolute truncation error below 10⁻⁵, and justify the choice with an error estimate.

Multiple Choice Questions with Answers

Question 1

Which condition is necessary for the convergence of ∑a_n?

A. lim_n→∞a_n=1

B. lim_n→∞a_n=0

C. Every term must be positive

D. The terms must be increasing

Correct Answer: B. lim_n→∞a_n=0

Question 2

The series ∑_n=1^∞1/n^p converges when:

A. p < 0

B. p = 0

C. p > 1

D. 0 < p ≤ 1

Correct Answer: C. p > 1

Question 3

For the ratio test, a limit L < 1 implies:

A. Divergence

B. Conditional convergence only

C. Absolute convergence

D. The test is inconclusive

Correct Answer: C. Absolute convergence

Question 4

The interval of convergence of ∑_n=1^∞(x−2)ⁿ/n is:

A. (1,3)

B. [1,3)

C. (1,3]

D. [1,3]

Correct Answer: B. [1,3)

Question 5

The Maclaurin series is a Taylor series centered at:

A. 1

B. −1

C. An arbitrary c

D. 0

Correct Answer: D. 0

Question 6

Using the cubic approximation sin x≈x−x³/6, the next omitted term in magnitude at x = 0.2 is approximately:

A. 2.667 × 10⁻⁶

B. 3.333 × 10⁻⁴

C. 6.667 × 10⁻³

D. 2.000 × 10⁻¹

Correct Answer: A. 2.667 × 10⁻⁶

Question 7

If a power series has radius of convergence R, what generally happens when |x−c|>R?

A. It converges absolutely

B. It diverges

C. It becomes a finite polynomial

D. It converges only conditionally

Correct Answer: B. It diverges

Question 8

A conditionally convergent series is one that:

A. Has all positive terms

B. Diverges but has bounded terms

C. Converges while its absolute-value series diverges

D. Has a ratio-test limit equal to zero

Correct Answer: C. Converges while its absolute-value series diverges

Question 9

In a first-order multivariable Taylor model, the Jacobian represents:

A. Second derivatives only

B. First partial derivatives arranged in a matrix

C. The series remainder

D. The radius of convergence

Correct Answer: B. First partial derivatives arranged in a matrix

Question 10

For e^x, a valid bound on the fourth-derivative magnitude over [0,0.1] is:

A. 0

B. 1

C. At most e⁰·¹

D. At most e⁻⁰·¹

Correct Answer: C. At most e⁰·¹