An infinite geometric series diverges when the sum of its terms does not approach a finite value; instead, it grows infinitely large (positive or negative) or oscillates without settling on a specific number.
Understanding Divergence
In simpler terms, if you keep adding the terms of a geometric series and the total gets bigger and bigger without limit, or if the sum jumps around without ever getting close to a single number, then the series diverges.
Key Concepts
- Geometric Series: A series where each term is multiplied by a constant ratio (r) to get the next term. The general form is a + ar + ar2 + ar3 + ...
- Convergence: A series converges if the sum of its terms approaches a finite value as the number of terms increases infinitely.
- Divergence: A series diverges if it does not converge.
Condition for Divergence
An infinite geometric series diverges if the absolute value of the common ratio (r) is greater than or equal to 1 ( |r| ≥ 1 ). This is because the terms either stay the same size or get larger and larger, so the sum will never approach a finite limit.
Examples of Divergent Geometric Series
- r > 1: The series 1 + 2 + 4 + 8 + ... diverges because each term is larger than the previous one, causing the sum to grow without bound.
- r = 1: The series 1 + 1 + 1 + 1 + ... diverges because the sum increases by 1 with each term added.
- r = -1: The series 1 - 1 + 1 - 1 + ... diverges because the sum oscillates between 0 and 1, never settling on a single value.
- r < -1: The series 1 - 2 + 4 - 8 + ... diverges because the terms alternate in sign and increase in magnitude, causing the sum to oscillate with increasing amplitude.
Formula for the Sum of an Infinite Geometric Series
The formula for the sum (S) of an infinite geometric series is:
S = a / (1 - r)
where:
- a is the first term
- r is the common ratio
Important: This formula only works when |r| < 1 (i.e., the series converges). If |r| ≥ 1, the series diverges, and the sum is not a finite number.
In Summary
When an infinite geometric series diverges, it means that its sum does not approach a finite value. This happens when the absolute value of the common ratio is greater than or equal to 1, causing the terms to either grow without bound or oscillate indefinitely, preventing the sum from converging to a specific number.
