Unit 3 — Sequence, Series and Fourier Series

This unit introduces sequences, infinite series, convergence tests, and Fourier series. You will learn the fundamental formulas used to determine convergence and represent periodic functions using trigonometric series.

Part A — Sequence and Series

1. Sequence and Series

A sequence is an ordered list of numbers:

a_1, a_2, a_3, \ldots, a_n, \ldots

It can be represented as:

\{a_n\}_{n=1}^{\infty}

The $n$ -th term of a sequence is denoted by $a_n$ .

Limit of a Sequence

A sequence $\{a_n\}$ converges to $L$ if:

\lim_{n \to \infty} a_n = L

A sequence is divergent if the limit does not exist or is not finite.

2. Infinite Series

An infinite series is the sum of the terms of a sequence:

a_1 + a_2 + a_3 + \cdots

\sum_{n=1}^{\infty} a_n

The $n$ -th partial sum is:

S_n = \sum_{k=1}^{n} a_k

The series converges to $S$ if:

\lim_{n \to \infty} S_n = S

Therefore:

\sum_{n=1}^{\infty} a_n = S

when the limit exists and is finite.

Necessary Condition for Convergence

If $\displaystyle\sum_{n=1}^{\infty} a_n$ converges, then necessarily:

\lim_{n \to \infty} a_n = 0

\lim_{n \to \infty} a_n \neq 0

then the series diverges.

3. Common Series

Geometric Series

A geometric series has the form:

a + ar + ar^2 + ar^3 + \cdots

\sum_{n=0}^{\infty} ar^n

The series converges when:

|r| < 1

Its sum is:

S = \frac{a}{1 - r}

If $|r| \geq 1$ , the infinite geometric series diverges.

Harmonic Series

The harmonic series is:

\sum_{n=1}^{\infty} \frac{1}{n}

It is divergent.

$p$ -Series

A $p$ -series is:

\sum_{n=1}^{\infty} \frac{1}{n^p}

It converges when:

p > 1

and diverges when:

p \leq 1

4. Convergence of Series

A series $\displaystyle\sum_{n=1}^{\infty} a_n$ is convergent if its partial sums approach a finite limit:

\lim_{n \to \infty} \sum_{k=1}^{n} a_k = S

Otherwise, the series is divergent.

The main convergence tests studied in this unit are:

Ratio Test
D'Alembert's Test
Raabe's Test
Comparison Test

5. Ratio Test

For the series $\displaystyle\sum_{n=1}^{\infty} a_n$ , calculate:

L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|

Case 1 — $L < 1$

L < 1 \quad \Rightarrow \quad \text{Series converges absolutely}

Case 2 — $L > 1$

L > 1 \quad \Rightarrow \quad \text{Series diverges}

Case 3 — $L = 1$

L = 1 \quad \Rightarrow \quad \text{Test is inconclusive}

6. D'Alembert's Test

D'Alembert's test is also known as the ratio test.

For $\displaystyle\sum_{n=1}^{\infty} a_n$ , evaluate:

L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|

Then:

L < 1 \quad \Rightarrow \quad \text{Convergent}

L > 1 \quad \Rightarrow \quad \text{Divergent}

L = 1 \quad \Rightarrow \quad \text{Inconclusive}

7. Raabe's Test

For the positive-term series $\displaystyle\sum_{n=1}^{\infty} a_n$ , Raabe's test uses:

R = \lim_{n \to \infty} n \left( \frac{a_n}{a_{n+1}} - 1 \right)

Case 1 — $R > 1$

R > 1 \quad \Rightarrow \quad \text{Series converges}

Case 2 — $R < 1$

R < 1 \quad \Rightarrow \quad \text{Series diverges}

Case 3 — $R = 1$

R = 1 \quad \Rightarrow \quad \text{Test is inconclusive}

Raabe's test is useful when the ratio test gives:

\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 1

8. Comparison Test

Suppose

0 \leq a_n \leq b_n

for sufficiently large $n$ .

Convergence

If $\displaystyle\sum_{n=1}^{\infty} b_n$ converges, then:

\sum_{n=1}^{\infty} a_n \quad \text{also converges}

Divergence

0 \leq b_n \leq a_n

and $\displaystyle\sum_{n=1}^{\infty} b_n$ diverges, then:

\sum_{n=1}^{\infty} a_n \quad \text{also diverges}

Limit Comparison Test

For positive sequences $a_n$ and $b_n$ , calculate:

L = \lim_{n \to \infty} \frac{a_n}{b_n}

If $0 < L < \infty$ , then:

\sum a_n \quad \text{and} \quad \sum b_n \quad \text{have the same behavior}

(That means either both converge or both diverge.)

Part B — Fourier Series

9. Fourier Series

A periodic function can be represented as a Fourier series containing sine and cosine terms.

For a function with period $2L$ :

f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[ a_n \cos\left(\frac{n\pi x}{L}\right) + b_n \sin\left(\frac{n\pi x}{L}\right) \right]

The coefficients are called Fourier coefficients.

10. Fourier Coefficients

The constant coefficient is:

a_0 = \frac{1}{L} \int_{-L}^{L} f(x)\, dx

The cosine coefficients are:

a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right)\, dx

The sine coefficients are:

b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right)\, dx

Therefore, the complete Fourier series is:

f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[ a_n \cos\left(\frac{n\pi x}{L}\right) + b_n \sin\left(\frac{n\pi x}{L}\right) \right]

11. Fourier Series on $(-\pi, \pi)$

For a function with period $2\pi$ , the Fourier series becomes:

f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos nx + b_n \sin nx)

The coefficients are:

a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x)\, dx

a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos nx\, dx

b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin nx\, dx

12. Fourier Series of an Even Function

f(-x) = f(x)

then $f(x)$ is an even function.

For an even function:

b_n = 0

Therefore, the Fourier series contains only cosine terms:

f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos nx

The coefficients can be calculated using:

a_0 = \frac{2}{\pi} \int_0^{\pi} f(x)\, dx

a_n = \frac{2}{\pi} \int_0^{\pi} f(x) \cos nx\, dx

13. Fourier Series of an Odd Function

f(-x) = -f(x)

then $f(x)$ is an odd function.

For an odd function:

a_0 = 0, \qquad a_n = 0

Therefore, the Fourier series contains only sine terms:

f(x) = \sum_{n=1}^{\infty} b_n \sin nx

The coefficients can be calculated using:

b_n = \frac{2}{\pi} \int_0^{\pi} f(x) \sin nx\, dx

14. Half-Range Fourier Sine Series

Suppose $f(x)$ is defined on $0 < x < L$ .

Its half-range Fourier sine series is:

f(x) = \sum_{n=1}^{\infty} b_n \sin\left(\frac{n\pi x}{L}\right)

where

b_n = \frac{2}{L} \int_0^L f(x) \sin\left(\frac{n\pi x}{L}\right)\, dx

This represents an odd extension of the function over the interval $(-L, L)$ .

15. Half-Range Fourier Cosine Series

For a function defined on $0 < x < L$ , the half-range Fourier cosine series is:

f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos\left(\frac{n\pi x}{L}\right)

where

a_0 = \frac{2}{L} \int_0^L f(x)\, dx

a_n = \frac{2}{L} \int_0^L f(x) \cos\left(\frac{n\pi x}{L}\right)\, dx

This represents an even extension of the function over $(-L, L)$ .

16. Fourier Series of a Function with Period $2L$

For a periodic function with $f(x + 2L) = f(x)$ , the Fourier series is:

f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[ a_n \cos\left(\frac{n\pi x}{L}\right) + b_n \sin\left(\frac{n\pi x}{L}\right) \right]

where

a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right)\, dx

b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right)\, dx

17. Complex Form of Fourier Series

A Fourier series can also be represented using complex exponentials:

f(x) = \sum_{n=-\infty}^{\infty} c_n e^{inx}

where

c_n = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) e^{-inx}\, dx

Using Euler's formula:

e^{inx} = \cos nx + i\sin nx

e^{-inx} = \cos nx - i\sin nx

the complex form can be related to the sine-cosine form.

18. Important Fourier Series Formulas

General Fourier Series

f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos nx + b_n \sin nx)

Constant Coefficient

a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x)\, dx

Cosine Coefficient

a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos nx\, dx

Sine Coefficient

b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin nx\, dx

Half-Range Sine Coefficient

b_n = \frac{2}{L} \int_0^L f(x) \sin\left(\frac{n\pi x}{L}\right)\, dx

Half-Range Cosine Coefficient

a_n = \frac{2}{L} \int_0^L f(x) \cos\left(\frac{n\pi x}{L}\right)\, dx

Summary of Convergence Tests

Necessary Test

\lim_{n \to \infty} a_n

Must be equal to $0$ for convergence.

Ratio Test (D'Alembert's Test)

L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|

$L < 1$ → Convergent
$L > 1$ → Divergent
$L = 1$ → Inconclusive

Raabe's Test

R = \lim_{n \to \infty} n \left( \frac{a_n}{a_{n+1}} - 1 \right)

$R > 1$ → Convergent
$R < 1$ → Divergent
$R = 1$ → Inconclusive

Comparison Test

0 \leq a_n \leq b_n

Compare with a known convergent or divergent series.

What You'll Learn

By completing this unit, you will be able to:

Understand sequences and infinite series.
Calculate limits of sequences.
Determine whether a series converges or diverges.
Apply the necessary condition for convergence.
Identify and use geometric series.
Apply the $p$ -series criterion.
Apply the Ratio Test.
Apply D'Alembert's Test.
Apply Raabe's Test.
Apply the Comparison Test.
Understand Fourier series representation.
Calculate Fourier coefficients.
Construct Fourier sine and cosine series.
Identify even and odd functions.
Construct half-range Fourier sine series.
Construct half-range Fourier cosine series.
Represent periodic functions using Fourier series.

Key Takeaway

This unit develops the mathematical tools needed to analyze infinite sequences and series and to represent periodic functions using Fourier series. These concepts are fundamental in engineering mathematics, signal processing, vibration analysis, and many other engineering applications.

Unit 3 — Sequence, Series and Fourier Series

Part A — Sequence and Series

1. Sequence and Series

Limit of a Sequence

2. Infinite Series

Necessary Condition for Convergence

3. Common Series

Geometric Series

Harmonic Series

ppp-Series

4. Convergence of Series

5. Ratio Test

Case 1 — L<1L < 1L<1

Case 2 — L>1L > 1L>1

Case 3 — L=1L = 1L=1

6. D'Alembert's Test

7. Raabe's Test

Case 1 — R>1R > 1R>1

Case 2 — R<1R < 1R<1

Case 3 — R=1R = 1R=1

8. Comparison Test

Convergence

Divergence

Limit Comparison Test

Part B — Fourier Series

9. Fourier Series

10. Fourier Coefficients

11. Fourier Series on (−π,π)(-\pi, \pi)(−π,π)

12. Fourier Series of an Even Function

13. Fourier Series of an Odd Function

14. Half-Range Fourier Sine Series

15. Half-Range Fourier Cosine Series

16. Fourier Series of a Function with Period 2L2L2L

17. Complex Form of Fourier Series

18. Important Fourier Series Formulas

General Fourier Series

Constant Coefficient

Cosine Coefficient

Sine Coefficient

Half-Range Sine Coefficient

Half-Range Cosine Coefficient

Summary of Convergence Tests

Necessary Test

Ratio Test (D'Alembert's Test)

Raabe's Test

Comparison Test

What You'll Learn

Key Takeaway

$p$ -Series

Case 1 — $L < 1$

Case 2 — $L > 1$

Case 3 — $L = 1$

Case 1 — $R > 1$

Case 2 — $R < 1$

Case 3 — $R = 1$

11. Fourier Series on $(-\pi, \pi)$

16. Fourier Series of a Function with Period $2L$