11.1 Introduction to sequences and series

Sequences and series are fundamental concepts in calculus, forming the backbone of many mathematical models. They're like building blocks, allowing us to represent complex patterns and relationships in a simple, ordered way.

Understanding sequences and series is crucial for grasping more advanced calculus topics. They're used in everything from financial calculations to physics equations, making them essential tools for problem-solving in various fields.

Sequences and their Properties

Defining Sequences

• A sequence is an ordered list of numbers, typically represented by a function with a domain consisting of the natural numbers
  • Can be finite, consisting of a fixed number of terms (1, 2, 3, 4, 5)
  • Can be infinite, continuing indefinitely (1, 2, 3, 4, 5, ...)
• The individual numbers in a sequence are called terms, often denoted by a subscript indicating their position
  • Examples: $a_1$, $a_2$, $a_3$, ...
• The general term of a sequence, usually denoted as $a_n$, represents the nth term and is often expressed as a function of n

Key Properties of Sequences

• Include the first term ($a_1$), which is the initial value of the sequence
• Common difference between consecutive terms in arithmetic sequences
  • Represents the constant amount by which each term differs from the previous one
• Common ratio between consecutive terms in geometric sequences
  • Represents the constant factor by which each term is multiplied to obtain the next term

Arithmetic vs Geometric Sequences

Arithmetic Sequences

• An arithmetic sequence is a sequence in which the difference between any two consecutive terms is constant, called the common difference (d)
  • Example: 2, 5, 8, 11, 14, ... (common difference is 3)
• The general term of an arithmetic sequence is given by $a_n = a_1 + (n - 1)d$
  • $a_1$ is the first term
  • d is the common difference
• Arithmetic sequences have a linear growth pattern, meaning the terms increase or decrease by a constant amount

Geometric Sequences

• A geometric sequence is a sequence in which the ratio between any two consecutive terms is constant, called the common ratio (r)
  • Example: 2, 6, 18, 54, 162, ... (common ratio is 3)
• The general term of a geometric sequence is given by $a_n = a_1 \times r^{n-1}$
  • $a_1$ is the first term
  • r is the common ratio
• Geometric sequences exhibit exponential growth or decay depending on the value of the common ratio
  • If r > 1, the sequence grows exponentially (doubles, triples, etc.)
  • If 0 < r < 1, the sequence decays exponentially (halves, thirds, etc.)

Calculating Sequence Terms

Arithmetic Sequence Terms

• To find the nth term of an arithmetic sequence, use the formula $a_n = a_1 + (n - 1)d$
  • Substitute the values for $a_1$ (first term), n (position of the desired term), and d (common difference)
  • Example: For the sequence 3, 7, 11, 15, ..., find the 10th term
    • $a_1 = 3$, $d = 4$, $n = 10$
    • $a_{10} = 3 + (10 - 1)4 = 3 + 36 = 39$

Geometric Sequence Terms

• To find the nth term of a geometric sequence, use the formula $a_n = a_1 \times r^{n-1}$
  • Substitute the values for $a_1$ (first term), n (position of the desired term), and r (common ratio)
  • Example: For the sequence 2, 6, 18, 54, ..., find the 6th term
    • $a_1 = 2$, $r = 3$, $n = 6$
    • $a_6 = 2 \times 3^{5} = 2 \times 243 = 486$

Identifying Sequence Patterns

• When given the first few terms of a sequence, identify the pattern to determine whether it is an arithmetic or geometric sequence
  • If the difference between consecutive terms is constant, it is an arithmetic sequence
  • If the ratio between consecutive terms is constant, it is a geometric sequence
• Once the pattern is identified, use the appropriate formula to calculate the desired term

Sequence Convergence vs Divergence

Defining Convergence and Divergence

• A sequence is said to converge if it approaches a specific finite value, called the limit, as n approaches infinity
  • Example: The sequence $\frac{1}{n}$ (1, $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, ...) converges to 0
• If a sequence does not approach a specific finite value, it is said to diverge
  • Example: The sequence $n$ (1, 2, 3, 4, ...) diverges to infinity

Convergence and Divergence of Arithmetic Sequences

• If the common difference is zero, the sequence converges to the first term
  • Example: 5, 5, 5, 5, ... converges to 5
• If the common difference is non-zero, the sequence diverges
  • Example: 1, 4, 7, 10, ... diverges to infinity

Convergence and Divergence of Geometric Sequences

• If the absolute value of the common ratio is less than 1 ($|r| < 1$), the sequence converges to 0
  • Example: 1, $\frac{1}{2}$, $\frac{1}{4}$, $\frac{1}{8}$, ... converges to 0
• If the absolute value of the common ratio is greater than or equal to 1 ($|r| \geq 1$), the sequence diverges
  • Example: 2, 4, 8, 16, ... diverges to infinity

Determining Convergence or Divergence

• To determine the convergence or divergence of a sequence, examine its behavior as n approaches infinity by calculating the limit of the general term
  • If the limit exists and is finite, the sequence converges
  • If the limit does not exist or is infinite, the sequence diverges
• Use the properties of arithmetic and geometric sequences to help determine convergence or divergence