geometric series

Thân thiện

A student writes out a geometric series on the chalkboard.

Definition

1. Noun:
   • A series of terms where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio, expressed as a sum: A geometric series is the summation of the terms of a geometric progression. It is typically written in the form ( a + ar + ar^2 + ar^3 + ... + ar^{n-1} ), where ( a ) is the first term and ( r ) is the common ratio.

Usage Examples

• Noun:
  • The infinite geometric series 1 + 1/2 + 1/4 + 1/8 + ... converges to 2.
  • To find the total, you need to calculate the sum of the finite geometric series.

Advanced Usage

• Convergence of an infinite geometric series: An infinite geometric series converges to the sum ( \frac{a}{1-r} ) if the absolute value of the common ratio ( |r| < 1 ). If ( |r| \geq 1 ), the series diverges.
  • The convergence of the geometric series is a fundamental concept in calculus.

Variants and Related Words

• Geometric progression (n): The sequence of numbers from which a geometric series is formed. Example:
• Common ratio (n): The constant factor between consecutive terms of a geometric progression or series.
• Series (n): The sum of the terms of a sequence.

Synonyms

• Geometric sum: A sum resulting from a geometric progression.

Related Phrases and Concepts

• Sum of a geometric series: The total value obtained by adding all terms. For a finite series with ( n ) terms, the sum is ( S_n = a(1 - r^n) / (1 - r) ) for ( r \neq 1 ).
  • The formula for the sum of a geometric series is essential for solving many financial and growth problems.
• Divergent geometric series: A geometric series that does not converge to a finite limit, typically when ( |r| \geq 1 ).
  • The series 2 + 4 + 8 + 16 + ... is a divergent geometric series.

Học thuật

Noun

1. a geometric progression written as a sum