WhizMath

Introduction: The Rhythm of Numbers

Welcome to Whizmath, where we reveal the hidden patterns and powerful sums within numbers! Today, we delve into the fascinating world of "Sequences and Series"—two interconnected concepts that describe ordered lists of numbers and their sums. These mathematical tools are not just abstract ideas; they are fundamental to understanding phenomena in finance (compound interest), biology (population growth), computer science (algorithms), and physics (wave patterns).

In this comprehensive lesson, we will explore the definitions, types, and properties of sequences and series. You'll learn how to identify patterns, calculate specific terms, and find sums of various number progressions. Prepare to discover the elegance of numerical order and the power of infinite sums!

Chapter 1: Sequences – Ordered Lists of Numbers

A sequence is an ordered list of numbers. Each number in the sequence is called a term. We often denote the terms of a sequence as $a_1, a_2, a_3, \dots, a_n, \dots$, where $a_n$ is the $n$-th term.

1.1 Arithmetic Sequences: Constant Difference

An arithmetic sequence is a sequence where the difference between consecutive terms is constant. This constant difference is called the common difference, denoted by $d$.

General Term (n-th term) of an Arithmetic Sequence:

$a_n = a_1 + (n-1)d$

Where:

• $a_n$ is the $n$-th term
• $a_1$ is the first term
• $n$ is the term number
• $d$ is the common difference

Example 1.1.1:

Consider the sequence: $2, 5, 8, 11, \dots$

• $a_1 = 2$
• Common difference $d = 5 - 2 = 3$ (or $8 - 5 = 3$)
• To find the 10th term ($a_{10}$): $a_{10} = 2 + (10-1)3 = 2 + 9 \times 3 = 2 + 27 = 29$

1.2 Geometric Sequences: Constant Ratio

A geometric sequence is a sequence where the ratio between consecutive terms is constant. This constant ratio is called the common ratio, denoted by $r$.

General Term (n-th term) of a Geometric Sequence:

$a_n = a_1 r^{n-1}$

Where:

• $a_n$ is the $n$-th term
• $a_1$ is the first term
• $n$ is the term number
• $r$ is the common ratio

Example 1.2.1:

Consider the sequence: $3, 6, 12, 24, \dots$

• $a_1 = 3$
• Common ratio $r = \frac{6}{3} = 2$ (or $\frac{12}{6} = 2$)
• To find the 7th term ($a_7$): $a_7 = 3 \cdot 2^{7-1} = 3 \cdot 2^6 = 3 \cdot 64 = 192$

Chapter 2: Series – The Sum of Sequences

A series is the sum of the terms of a sequence. If a sequence is $a_1, a_2, a_3, \dots$, then the corresponding series is $a_1 + a_2 + a_3 + \dots$.

2.1 Finite vs. Infinite Series

• Finite Series: The sum of a finite number of terms. For example, the sum of the first 5 terms of a sequence.
• Infinite Series: The sum of an infinite number of terms. These can either converge (sum to a finite value) or diverge (sum to infinity).

2.2 Summation Notation ($\Sigma$)

We use summation notation (or sigma notation) to represent series concisely.

$\sum_{k=1}^{n} a_k = a_1 + a_2 + a_3 + \dots + a_n$

Where:

• $\Sigma$ (capital sigma) means "sum".
• $k$ is the index of summation (dummy variable).
• $1$ is the lower limit of summation (starting value of $k$).
• $n$ is the upper limit of summation (ending value of $k$).
• $a_k$ is the formula for the $k$-th term.

Example 2.2.1:

$\sum_{k=1}^{4} (2k + 1) = (2(1)+1) + (2(2)+1) + (2(3)+1) + (2(4)+1)$

$ = 3 + 5 + 7 + 9 = 24$

Chapter 3: Arithmetic Series – Summing a Constant Difference

An arithmetic series is the sum of the terms of an arithmetic sequence.

3.1 Sum of a Finite Arithmetic Series

The sum of the first $n$ terms of an arithmetic series, denoted by $S_n$, can be found using two main formulas:

Formula 1 (using first and last term):

$S_n = \frac{n}{2}(a_1 + a_n)$

Formula 2 (using first term and common difference):

$S_n = \frac{n}{2}(2a_1 + (n-1)d)$

Example 3.1.1:

Find the sum of the first 10 terms of the arithmetic sequence: $2, 5, 8, 11, \dots$

• $a_1 = 2$, $d = 3$, $n = 10$.
• First, find $a_{10}$: $a_{10} = 2 + (10-1)3 = 2 + 27 = 29$.
• Now, use $S_n = \frac{n}{2}(a_1 + a_n)$:
• $S_{10} = \frac{10}{2}(2 + 29) = 5(31) = 155$

Example 3.1.2:

Find the sum of the first 20 even numbers (starting from 2).

• This is an arithmetic sequence: $2, 4, 6, \dots$
• $a_1 = 2$, $d = 2$, $n = 20$.
• Using $S_n = \frac{n}{2}(2a_1 + (n-1)d)$:
• $S_{20} = \frac{20}{2}(2(2) + (20-1)2)$
• $S_{20} = 10(4 + 19 \times 2)$
• $S_{20} = 10(4 + 38) = 10(42) = 420$

Chapter 4: Geometric Series – Summing a Constant Ratio

A geometric series is the sum of the terms of a geometric sequence.

4.1 Sum of a Finite Geometric Series

The sum of the first $n$ terms of a geometric series, denoted by $S_n$, is given by:

$S_n = \frac{a_1(1 - r^n)}{1 - r}$, where $r \neq 1$

Example 4.1.1:

Find the sum of the first 6 terms of the geometric sequence: $3, 6, 12, 24, \dots$

• $a_1 = 3$, $r = 2$, $n = 6$.
• $S_6 = \frac{3(1 - 2^6)}{1 - 2} = \frac{3(1 - 64)}{-1} = \frac{3(-63)}{-1} = 189$

4.2 Sum of an Infinite Geometric Series

An infinite geometric series can converge (have a finite sum) only if the absolute value of the common ratio is less than 1 (i.e., $|r| < 1$). If $|r| \geq 1$, the series diverges (its sum approaches infinity or oscillates).

Sum of an Infinite Geometric Series (when $|r| < 1$):

$S = \frac{a_1}{1 - r}$

Example 4.2.1:

Find the sum of the infinite geometric series: $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$

• $a_1 = 1$
• $r = \frac{1}{2}$
• Since $|r| = |\frac{1}{2}| < 1$, the series converges.
• $S = \frac{1}{1 - \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2$

Example 4.2.2:

Does the series $2 - 4 + 8 - 16 + \dots$ converge or diverge? If it converges, find its sum.

• $a_1 = 2$
• $r = \frac{-4}{2} = -2$
• Since $|r| = |-2| = 2 \geq 1$, the series diverges.

Chapter 5: Special Series and Convergence/Divergence

Beyond arithmetic and geometric series, there are many other types of series. Determining whether an infinite series converges or diverges is a central topic in calculus.

5.1 Harmonic Series

The harmonic series is defined as:

$\sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots$

Despite the terms getting smaller, the harmonic series diverges (its sum goes to infinity). This is a classic counter-intuitive result.

5.2 P-Series

A p-series is a series of the form:

$\sum_{n=1}^{\infty} \frac{1}{n^p} = 1 + \frac{1}{2^p} + \frac{1}{3^p} + \frac{1}{4^p} + \dots$

A p-series converges if $p > 1$ and diverges if $p \leq 1$. The harmonic series is a p-series with $p=1$, which confirms its divergence.

5.3 Introduction to Convergence Tests (Briefly)

For more complex infinite series, various tests are used to determine convergence or divergence without necessarily finding the sum. These tests are typically covered in calculus courses:

• Divergence Test (n-th Term Test): If $\lim_{n \to \infty} a_n \neq 0$, then the series $\sum a_n$ diverges. (If the limit is 0, the test is inconclusive).
• Integral Test: Compares the series to an improper integral.
• Comparison Tests (Direct and Limit): Compares the series to a known convergent or divergent series.
• Ratio Test: Uses the ratio of consecutive terms, often effective for series involving factorials or exponentials.
• Root Test: Uses the $n$-th root of the absolute value of the terms.
• Alternating Series Test: For series with alternating signs.

Chapter 6: Real-World Applications of Sequences and Series

Sequences and series are powerful mathematical tools with diverse applications:

• Finance:
  • Compound Interest: The growth of money with compound interest follows a geometric sequence.
  • Loan Payments/Annuities: Calculating future values of investments or present values of loans often involves sums of geometric series.
• Population Growth/Decay: Many models of population change can be described using geometric sequences.
• Physics:
  • Falling Objects: The distance an object falls in successive seconds forms an arithmetic sequence (ignoring air resistance).
  • Wave Phenomena: Describing wave patterns and oscillations often involves series (e.g., Fourier series).
• Computer Science:
  • Algorithms: Analyzing the efficiency of algorithms often involves summing series to determine time complexity.
  • Data Structures: Understanding how data is stored and accessed can involve sequential thinking.
• Biology: Modeling bacterial growth, radioactive decay, or drug concentrations in the body.
• Engineering: Designing structures, analyzing electrical circuits, and signal processing.
• Art and Architecture: The Fibonacci sequence (a type of recursive sequence) appears in natural patterns and is used in design.

Practice Problems (with Solutions)