Compute the n-th term aₙ = a · rⁿ⁻¹, finite partial sum Sₙ, and infinite convergent series S_∞ = a / (1 - r).
Sample Elements:
2, 6, 18, 54, 162, 486 ...
GP Results
Term #6
N-th Term (aₙ)486
Finite Sum (Sₙ)728
Infinite Sum (S_∞)Diverges (|r| ≥ 1)
Visual Guides & Sequences Architecture
Authoritative reference charts illustrating AP and GP sequence progressions, infinite series convergence, and mean inequalities.
AP, GP & HP Sequences Formulations
Mathematical closed-form equations for arithmetic progressions, geometric decay, and the AM ≥ GM ≥ HM inequality.
Series Summations & Gauss Laws
Formulas for sums of natural numbers, squares, and cubes derived from Gauss summation principles.
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]
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Compute the n-th term aₙ = a · rⁿ⁻¹, finite partial sum Sₙ, and infinite convergent series S_∞ = a / (1 - r).
Sample Elements:
2, 6, 18, 54, 162, 486 ...
GP Results
Term #6
N-th Term (aₙ)486
Finite Sum (Sₙ)728
Infinite Sum (S_∞)Diverges (|r| ≥ 1)
Visual Guides & Sequences Architecture
Authoritative reference charts illustrating AP and GP sequence progressions, infinite series convergence, and mean inequalities.
AP, GP & HP Sequences Formulations
Mathematical closed-form equations for arithmetic progressions, geometric decay, and the AM ≥ GM ≥ HM inequality.
Series Summations & Gauss Laws
Formulas for sums of natural numbers, squares, and cubes derived from Gauss summation principles.
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:
