Comprehensive Reference: Progressions, Mathematical Sequences, and Series Convergence
In pure mathematics and financial engineering, numerical progressions describe interest compounding, loan amortizations, radioactive half-life decay, and harmonic acoustics.
1. Infinite Geometric Series and Convergence
When the absolute value of the common ratio is strictly less than one ($|r| < 1$), higher powers of $r$ diminish to zero ($r^n \to 0$ as $n \to \infty$). Consequently, the finite sum formula simplifies into the famous infinite sum:
S_∞ = a / (1 - r) [strictly for |r| < 1]