What is a Geometric Series?
A geometric series is the sum of the terms of a geometric sequence. In a geometric sequence, each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. When these terms are added together, they form a geometric series.
Key Components
A geometric series is defined by its first term (a), its common ratio (r), and the number of terms (n). The common ratio 'r' is crucial; if the absolute value of r (|r|) is less than 1, the series converges to a finite sum, while if |r| is greater than or equal to 1, the series diverges (except for trivial cases).
A Practical Example
Consider the series 2 + 4 + 8 + 16 + ... Here, the first term (a) is 2, and the common ratio (r) is 2 (since each term is multiplied by 2 to get the next). If we sum the first 4 terms of this series, the result is 2 + 4 + 8 + 16 = 30.
Importance and Applications
Geometric series are widely used in finance for calculating loan repayments and compound interest, in physics for modeling phenomena like radioactive decay or oscillating systems, and in computer science for analyzing algorithms. They are also fundamental for understanding infinite sums in calculus and the concept of convergence.