Math Sequences and Series

Sequence and Series

Sequences

• A sequence is an ordered list of numbers, denoted by {a_n} where n is a positive integer.
• Each term in the sequence is denoted by a_n, where n is the term number.
• A sequence can be finite (has a last term) or infinite (has no last term).

Types of Sequences

• Arithmetic sequence: a sequence in which each term is obtained by adding a fixed constant to the previous term.
  • Example: 2, 5, 8, 11, ...
• Geometric sequence: a sequence in which each term is obtained by multiplying the previous term by a fixed constant.
  • Example: 2, 6, 18, 34, ...
• Harmonic sequence: a sequence in which each term is the reciprocal of an arithmetic sequence.
  • Example: 1, 1/2, 1/3, 1/4, ...

Series

• A series is the sum of the terms of a sequence.
• The sum of a series can be finite or infinite.
• A series can be represented as:
  • Σa_n = a_1 + a_2 + ... + a_n

Types of Series

• Arithmetic series: the sum of an arithmetic sequence.
  • Example: 2 + 5 + 8 + 11 + ...
• Geometric series: the sum of a geometric sequence.
  • Example: 2 + 6 + 18 + 34 + ...
• Convergent series: a series that has a finite sum.
  • Example: 1 + 1/2 + 1/4 + 1/8 + ...
• Divergent series: a series that has an infinite sum.
  • Example: 1 + 2 + 4 + 8 + ...

Formulas

• Arithmetic series formula: Σa_n = (n/2)(a_1 + a_n)
• Geometric series formula: Σa_n = a_1(1 - r^n) / (1 - r)
• Sum of an infinite geometric series: Σa_n = a_1 / (1 - r), |r| < 1

Sequences

• A sequence is an ordered list of numbers, denoted by {a_n}, where n is a positive integer.
• Each term in the sequence is denoted by a_n, where n is the term number.
• Sequences can be finite or infinite.

Types of Sequences

• Arithmetic sequence: each term is obtained by adding a fixed constant to the previous term.
• Geometric sequence: each term is obtained by multiplying the previous term by a fixed constant.
• Harmonic sequence: each term is the reciprocal of an arithmetic sequence.

Series

• A series is the sum of the terms of a sequence.
• The sum of a series can be finite or infinite.
• A series can be represented as: Σa_n = a_1 + a_2 +...+ a_n.

Types of Series

• Arithmetic series: the sum of an arithmetic sequence.
• Geometric series: the sum of a geometric sequence.
• Convergent series: a series that has a finite sum.
• Divergent series: a series that has an infinite sum.

Formulas

• Arithmetic series formula: Σa_n = (n/2)(a_1 + a_n).
• Geometric series formula: Σa_n = a_1(1 - r^n) / (1 - r).
• Sum of an infinite geometric series: Σa_n = a_1 / (1 - r), |r| < 1.