Sequences, Series and Fourier Analysis

Introduction

Sequences and series provide mathematical methods for representing quantities that evolve step by step, accumulate through repeated contributions, or can be approximated by finite sums. Fourier analysis extends these ideas to represent periodic functions and signals as combinations of sinusoidal components. Together, they are fundamental to numerical computation, vibration analysis, signal processing, control engineering, robotics, and mechatronic system design.

A mechatronics engineer may use a sequence to describe sampled sensor data, a power series to approximate a nonlinear actuator model, and a Fourier series to identify dominant vibration frequencies in a rotating machine. The central questions are:

• Does a sequence approach a definite limiting value?
• Does an infinite series have a finite sum?
• On what interval does a power series converge?
• How can a periodic signal be decomposed into harmonics?
• What do Fourier coefficients reveal about energy and system behavior?

Learning Objectives

After studying this chapter, a student should be able to:

1. Define sequences, subsequences, boundedness, monotonicity, and convergence.
2. Apply standard tests for convergence and divergence of infinite series.
3. Determine the radius and interval of convergence of a power series.
4. Construct and use Taylor and Maclaurin series.
5. Derive real and complex Fourier series using orthogonality.
6. Apply Parseval’s relation to calculate signal power or mean-square value.
7. Interpret Fourier coefficients as a frequency spectrum.
8. Analyze a periodic engineering signal and recognize limitations of Fourier representations.

1. Sequences and Their Limits

A sequence is an ordered list of numbers, written as {a_n} or a₁,a₂,a₃,..., where n is usually a positive integer. A sequence may be defined explicitly, such as a_n = 1/n, or recursively, such as a_n+1=0.5a_n+1.

A sequence {a_n} converges to L if its terms become arbitrarily close to L as n becomes large. This is written lim_n→∞ a_n=L. Formally, for every ε>0, there exists an integer N such that |a_n-L|<ε whenever n≥N.

The formal definition is important because convergence concerns eventual behavior, not whether the first few terms are close to the limit.

Fundamental properties

If a_n→A and b_n→B, then, provided the expressions are defined,

If f is continuous at A, then f(a_n)→f(A). These limit laws permit algebraic simplification and substitution.

A sequence is bounded above if there is a number M such that a_n≤M for all n. It is bounded below if a_n≥m for some m. A sequence is bounded if it is bounded both above and below.

A sequence is monotone increasing if a_n+1≥a_n and monotone decreasing if a_n+1≤a_n. The monotone convergence theorem states:

Every increasing sequence bounded above converges. Every decreasing sequence bounded below converges.

This theorem is especially useful for recursively generated numerical algorithms.

Example 1: A recursively defined sequence

Let a₁=1 and a_n+1=0.5a_n+2. If the sequence converges to L, then taking limits on both sides gives

Thus 0.5L=2 and L=4. To verify the result, define e_n=a_n-4. Then

so e_n=(0.5)^n-1e₁→0. Therefore a_n→4.

A sequence can be bounded but divergent. For example, a_n=(-1)ⁿ remains between −1 and 1 but oscillates and has no limit.

2. Infinite Series

An infinite series is the formal sum

It is defined through its sequence of partial sums:

The series converges to S if S_N→S. Otherwise, it diverges.

Necessary condition for convergence

If ∑a_n converges, then

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

Geometric series

For common ratio r,

where a is the first term. If |r|≥1, the series diverges. The finite geometric sum is

Comparison tests

For nonnegative terms, if 0≤a_n≤b_n and ∑b_n converges, then ∑a_n converges. If 0≤b_n≤a_n and ∑b_n diverges, then ∑a_n diverges.

The limit comparison test is useful when a_n and b_n have similar asymptotic behavior. If

then ∑a_n and ∑b_n either both converge or both diverge.

Integral test

If f(x) is positive, continuous, and decreasing for x≥N, then ∑_n=N^∞f(n) and ∫_N^∞f(x)dx have the same convergence behavior.

For the p-series,

converges if and only if p>1.

Ratio and root tests

For a series with nonzero terms, the ratio test examines

If ρ<1, the series converges absolutely; if ρ>1 or the limit is infinite, it diverges; if ρ=1, the test is inconclusive.

The root test examines

The same conclusions apply. The root test is often effective for terms containing nth powers.

Alternating series

If b_n≥0, b_n+1≤b_n, and b_n→0, then

converges. Its remainder after N terms satisfies

Absolute convergence, ∑|a_n|<∞, implies convergence. Conditional convergence occurs when ∑a_n converges but ∑|a_n| diverges.

3. Power Series and Taylor Expansions

A power series centered at c has the form

Its convergence is described by a radius R. It converges absolutely when |x-c|<R, diverges when |x-c|>R, and requires separate testing at the endpoints x=c±R.

For coefficients that are not eventually zero, the ratio formula is

when the limit exists. Equivalently, the root test can be used.

Taylor and Maclaurin series

If f has derivatives of all required orders near c, its Taylor series is

A Maclaurin series is the special case c=0. The Taylor polynomial of degree N is

The Lagrange remainder is

for some ξ between x and c, assuming the required derivative exists on the interval.

Important Maclaurin expansions include

The last expansion is geometric and cannot be used outside its interval of convergence.

Example 2: Approximating an actuator nonlinearity

Suppose a displacement model contains f(x)=sin x, with x measured in radians. Near zero,

At x=0.2,

The exact value is approximately 0.1986693, demonstrating the usefulness of a short polynomial model for small-angle control calculations. The angle must be dimensionless in the mathematical series; radians are dimensionless in this sense.

4. Fourier Series

A Fourier series represents a periodic function as a sum of sines and cosines. Let f(t) have period T, and define the fundamental angular frequency

where T is measured in seconds and ω₀ in radians per second. The real Fourier series is

Under standard Dirichlet conditions—such as piecewise smoothness and a finite number of discontinuities in one period—the series converges to f(t) at points of continuity and to the midpoint of the one-sided limits at a jump.

The coefficients are

The integration interval can be any interval of length T. The coefficient a₀/2 is the average, or DC, value of the signal.

Orthogonality

Over any complete period, different sinusoidal harmonics are orthogonal:

For n≥1, the squared integrals of sine or cosine equal T/2. These identities isolate each coefficient when the Fourier series is multiplied by a selected basis function and integrated.

Symmetry shortcuts

If f(t) is even, then all b_n=0, and only cosine terms remain. If it is odd, then a₀=0 and all a_n=0. Half-wave symmetry, f(t+T/2)=-f(t), eliminates even harmonics.

Example 3: Symmetric square wave

Consider the periodic square wave of amplitude A and period 2π, defined over (−π,π) by f(x)=A for 0<x<π and f(x)=−A for −π<x<0. It is odd, so only sine terms occur. With period 2π,

Using odd symmetry,

Hence b_n=4A/(nπ) for odd n and b_n=0 for even n. Therefore,

The discontinuity produces overshoot near transitions, known as the Gibbs phenomenon. Increasing the number of harmonics narrows the region of error but does not eliminate the limiting overshoot at an ideal jump.

5. Complex Fourier Series

Using Euler’s identity,

where i²=−1, the Fourier series can be written

with complex coefficients

For a real-valued signal, c_−n=c_n^*, where the star denotes complex conjugation. The relationship with the real coefficients is

Complex notation is compact and particularly convenient in linear systems, spectral analysis, modulation, and vibration theory. The magnitude |c_n| indicates the strength of the nth harmonic, while its argument gives phase information.

6. Parseval’s Relation and Spectral Interpretation

Parseval’s relation connects the average square of a periodic signal with its Fourier coefficients. For the real series,

For the complex series,

If f(t) is a voltage across a resistance R_e in ohms, the average power is

and Parseval’s relation permits this power to be calculated from harmonic amplitudes. For mechanical vibration, the mean-square displacement, velocity, or acceleration can similarly be distributed among frequency components, although physical interpretation depends on which variable is measured.

A Fourier spectrum may be displayed as:

• a line spectrum for deterministic periodic signals;
• a magnitude spectrum showing |c_n| or sinusoidal amplitude;
• a phase spectrum showing harmonic phase;
• a power spectrum showing squared magnitude.

Frequency resolution depends on the observation duration. For a finite record of duration T_obs, the spacing of discrete Fourier frequencies is approximately

where ∆f is in hertz. This is a sampling and measurement consideration, not a replacement for the mathematical Fourier-series period.

7. Applied Case: Harmonic Diagnosis of a Rotating Machine

Background

The following is an illustrative engineering case. A motor-driven pump operates at 1,800 revolutions per minute. A vibration sensor measures radial acceleration, and the engineer suspects imbalance or misalignment.

Problem

The rotational frequency is

The measured periodic vibration contains a strong component at 30 Hz and a smaller component at 60 Hz. The task is to interpret the spectrum and recommend a diagnostic action.

Analysis

The fundamental rotational angular frequency is

A periodic vibration can be represented as

The 30 Hz line is the first rotational harmonic. A dominant first harmonic is commonly consistent with mass imbalance, although it is not conclusive by itself. A significant second harmonic can be associated with misalignment, looseness, or certain nonlinear effects. Phase and operating-condition data are needed before making a maintenance decision.

Approach and outcome

The engineer should:

1. Verify sensor calibration, mounting, sampling rate, and anti-alias filtering.
2. Record a sufficiently long time history and calculate the spectrum.
3. Compare radial and axial measurements.
4. Measure phase relative to a tachometer signal.
5. Inspect coupling alignment and balance only after confirming the evidence.

If the first harmonic dominates in the radial direction and phase remains stable, balancing is a reasonable first intervention. If the second harmonic is dominant in the axial direction, alignment inspection becomes more important. The Fourier spectrum narrows the diagnostic search, but it does not uniquely identify a fault.

Learning points

Fourier analysis transforms a time-domain problem into a frequency-domain problem. Harmonic frequencies should be compared with known machine speeds, gear-mesh frequencies, bearing characteristic frequencies, and structural natural frequencies. Safe engineering practice requires corroborating spectral evidence with physical inspection and operating data.

8. Numerical and Practical Considerations

A finite sampled record is commonly analyzed using the discrete Fourier transform (DFT):

where x_n is the nth sample and N is the number of samples. The inverse transform is

The fast Fourier transform (FFT) is an efficient algorithm for computing the DFT; it is not a different mathematical transform.

Important practical limitations include:

• Aliasing: Frequencies above half the sampling frequency can appear as false lower frequencies. An anti-alias filter and adequate sampling rate are required.
• Spectral leakage: Truncating a record that does not contain an integer number of periods spreads energy into neighboring bins. Window functions reduce leakage but alter amplitude calibration.
• Resolution: Longer records improve frequency resolution but may reduce the assumption of stationarity.
• Noise: Averaging spectra can improve estimates, but it can also conceal transient events.
• Discontinuities: Gibbs oscillations occur when a discontinuous waveform is represented by a finite harmonic sum.
• Nonstationarity: Fourier series describe periodic behavior; rapidly changing frequency or transient behavior may require short-time Fourier, wavelet, or order-tracking methods.

In industrial systems, spectral data may influence maintenance decisions. Engineers should preserve data provenance, record sensor and calibration information, protect operational data, and avoid presenting an automated classification as certain when the evidence is ambiguous. AI-based fault diagnosis can assist feature extraction and anomaly detection, but it should be validated against labeled failures and remain subject to engineering review.

Best-Practice Problem-Solving Procedure

1. Identify whether the object is a sequence, infinite series, power series, or periodic signal.
2. State the relevant assumptions, including smoothness, periodicity, sampling conditions, and units.
3. For convergence, test the term limit first, then choose a comparison, ratio, root, integral, or alternating-series test.
4. For power series, determine the radius and test endpoints separately.
5. For Fourier analysis, establish the period and fundamental frequency before calculating coefficients.
6. Use symmetry before integration whenever possible.
7. Check coefficients using limiting behavior, symmetry, units, or numerical reconstruction.
8. Interpret mathematical output in the physical context; a spectral peak is evidence, not automatically a diagnosis.

Chapter Summary

Sequences describe ordered values and converge when their terms approach a finite limit. Monotone bounded sequences converge, while boundedness alone does not guarantee convergence. Infinite series are defined through partial sums. Geometric, comparison, integral, ratio, root, and alternating-series tests provide systematic convergence criteria.

Power series have a radius of convergence and may require separate endpoint tests. Taylor and Maclaurin series approximate smooth functions locally, with the remainder providing an error estimate. Fourier series represent periodic signals using orthogonal sine and cosine functions. Their coefficients describe the DC component, harmonic amplitudes, and phases. Complex Fourier series provide a compact equivalent representation.

Parseval’s relation equates the mean square of a periodic signal with the sum of the mean-square contributions of its harmonics. In engineering, this supports power calculations, vibration diagnosis, and spectral interpretation. Practical Fourier analysis must account for sampling, aliasing, leakage, resolution, noise, and nonstationarity.

Key Terms

Sequence: An ordered list indexed by integers.

Limit: The value approached by a sequence or function.

Bounded sequence: A sequence confined between finite upper and lower bounds.

Monotone sequence: A sequence that never decreases or never increases.

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

Convergent series: A series whose partial sums approach a finite value.

Absolute convergence: Convergence of the series formed from absolute term values.

Geometric series: A series with a constant ratio between successive terms.

Power series: A series of powers of a variable about a center.

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

Taylor series: A power series formed from derivatives at a specified center.

Maclaurin series: A Taylor series centered at zero.

Fourier series: A representation of a periodic function using harmonics.

Fundamental frequency: The reciprocal of the period, in hertz.

Orthogonality: A zero inner product between distinct basis functions.

Harmonic: An integer multiple of a fundamental frequency.

Complex Fourier coefficient: A coefficient multiplying a complex exponential.

Parseval’s relation: An identity equating signal mean square with spectral contributions.

Gibbs phenomenon: Persistent oscillatory overshoot near a discontinuity.

Aliasing: Misrepresentation of high frequencies caused by insufficient sampling.

Review Questions

Short Answer Questions

1. What is the difference between a bounded sequence and a convergent sequence?
2. State the necessary term test for convergence of an infinite series.
3. Define the radius of convergence of a power series.
4. What physical quantity is represented by the Fourier coefficient a₀/2?
5. State one condition under which a Fourier series converges at a discontinuity.

Long Answer Questions

1. Explain and compare the ratio test, root test, and comparison test, including the cases in which each may be inconclusive.
2. Derive the real Fourier coefficients using the orthogonality of sine and cosine functions.
3. Explain Taylor-series approximation and derive the role of the Lagrange remainder in error estimation.
4. Derive the Fourier series of an odd square wave and explain why only odd sine harmonics occur.
5. Explain Parseval’s relation and discuss its relevance to electrical power and mechanical vibration.

Analytical / Application Questions

1. Determine the convergence of ∑_n=1^∞(3ⁿ)/(n·5ⁿ) and justify the test used.
2. A periodic signal has period 0.02 s. Determine its fundamental frequency and write the frequencies of its second and fifth harmonics.
3. A sampled vibration record lasts 4 s. Estimate its Fourier frequency spacing and discuss one trade-off associated with increasing the record duration.

Multiple Choice Questions with Answers

Question 1

Which condition is necessary for convergence of ∑a_n?

A. a_n→1

B. a_n→0

C. a_n is increasing

D. All terms are positive

Correct Answer: B. a_n→0

Question 2

The geometric series ∑_n=0^∞arⁿ converges when:

A. r=1

B. |r|>1

C. |r|<1

D. r is positive

Correct Answer: C. |r|<1

Question 3

For a power series centered at c, what must be done at x=c±R?

A. Assume divergence

B. Assume convergence

C. Test each endpoint separately

D. Change the center

Correct Answer: C. Test each endpoint separately

Question 4

Which Maclaurin expansion is correct?

A. cos x=∑xⁿ/n!

B. e^x=∑xⁿ/n!

C. sin x=∑x²ⁿ/(2n)!

D. 1/(1-x)=∑(−x)ⁿ

Correct Answer: B. e^x=∑xⁿ/n!

Question 5

If a periodic function is odd, which Fourier coefficients vanish?

A. All b_n

B. Only a₀

C. a₀ and all a_n

D. All coefficients

Correct Answer: C. a₀ and all a_n

Question 6

A signal with period 0.02 s has fundamental frequency:

A. 5 Hz

B. 20 Hz

C. 50 Hz

D. 200 Hz

Correct Answer: C. 50 Hz

Question 7

The complex Fourier coefficient is given by:

A. c_n=∫f(t)dt

B. c_n=(1/T)∫f(t)e^−inω₀tdt

C. c_n=(2/T)∫f(t)cos(nω₀t)dt

D. c_n=T∫f(t)dt

Correct Answer: B. c_n=(1/T)∫f(t)e^−inω₀tdt

Question 8

Parseval’s relation connects Fourier coefficients with the signal’s:

A. Period only

B. Mean square value

C. Maximum derivative

D. Sampling interval only

Correct Answer: B. Mean square value

Question 9

A dominant spectral peak at 30 Hz in a machine rotating at 1,800 rpm corresponds to:

A. The first rotational harmonic

B. The fifth rotational harmonic

C. A 1 Hz component

D. A DC component

Correct Answer: A. The first rotational harmonic

Question 10

For a 4 s sampled record, the approximate Fourier frequency spacing is:

A. 0.25 Hz

B. 2 Hz

C. 4 Hz

D. 8 Hz

Correct Answer: A. 0.25 Hz