Sequences in Mathematics

Sequence

• A sequence is an ordered list of numbers, often denoted as {a_n} or (a_n)
• Each term in the sequence is called an element or a term
• A sequence can be finite (having a fixed number of terms) or infinite (having an infinite number of terms)
• A sequence can be represented recursively or explicitly

Types of Sequences

• Arithmetic Sequence (AP):
  • Each term is obtained by adding a fixed constant to the previous term
  • General term: a_n = a_1 + (n-1)d, where d is the common difference
• Geometric Sequence (GP):
  • Each term is obtained by multiplying the previous term by a fixed constant
  • General term: a_n = a_1 * r^(n-1), where r is the common ratio
• Harmonic Sequence:
  • A sequence of reciprocals of positive integers
  • General term: a_n = 1/n
• Fibonacci Sequence:
  • A sequence of numbers in which each term is the sum of the two preceding terms
  • General term: a_n = a_(n-1) + a_(n-2), with a_1 = a_2 = 1

Series

• A series is the sum of the terms of a sequence
• A series can be finite or infinite
• A series can be denoted as Σa_n or ∑a_n

Types of Series

• Arithmetic Series:
  • The sum of an arithmetic sequence
  • General term: S_n = n/2 * (a_1 + a_n)
• Geometric Series:
  • The sum of a geometric sequence
  • General term: S_n = a_1 * (1 - r^n) / (1 - r), where |r| < 1
• Harmonic Series:
  • The sum of a harmonic sequence
  • General term: S_n = 1 + 1/2 + 1/3 + ... + 1/n
• Convergent and Divergent Series:
  • A convergent series has a finite sum as the number of terms increases without bound
  • A divergent series has an infinite sum or no sum at all

Sequence

• An ordered list of numbers, denoted as {a_n} or (a_n)
• Each term is an element or a term in the sequence
• Sequences can be finite or infinite
• Sequences can be represented recursively or explicitly

Types of Sequences

Arithmetic Sequence (AP)

• Each term is obtained by adding a fixed constant to the previous term
• General term: a_n = a_1 + (n-1)d, where d is the common difference

Geometric Sequence (GP)

• Each term is obtained by multiplying the previous term by a fixed constant
• General term: a_n = a_1 * r^(n-1), where r is the common ratio

Harmonic Sequence

• A sequence of reciprocals of positive integers
• General term: a_n = 1/n

Fibonacci Sequence

• A sequence of numbers where each term is the sum of the two preceding terms
• General term: a_n = a_(n-1) + a_(n-2), with a_1 = a_2 = 1

Series

• The sum of the terms of a sequence
• A series can be finite or infinite
• A series can be denoted as Σa_n or ∑a_n

Types of Series

Arithmetic Series

• The sum of an arithmetic sequence
• General term: S_n = n/2 * (a_1 + a_n)

Geometric Series

• The sum of a geometric sequence
• General term: S_n = a_1 * (1 - r^n) / (1 - r), where |r| < 1

Harmonic Series

• The sum of a harmonic sequence
• General term: S_n = 1 + 1/2 + 1/3 +...+ 1/n

Convergent and Divergent Series

• A convergent series has a finite sum as the number of terms increases without bound
• A divergent series has an infinite sum or no sum at all