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Free sequence convergence calculator & mathematical sequence calculator. Analyze sequence convergence, calculate limits, and apply convergence tests with step-by-step solutions. Our calculator supports various sequence types including harmonic, geometric, and alternating sequences.
Last updated: February 2, 2026
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Use standard notation: n for index, ^ for powers, / for division
The harmonic sequence 1/n diverges. Although the terms approach 0, the sum of all terms diverges to infinity.
This is a classic example of a sequence where terms approach 0 but the series diverges.
Sequence types
Harmonic, Geometric, Alternating
Analyzes convergence for harmonic, geometric, alternating, and rational sequences
Convergence tests
Ratio, Root, Comparison, Integral
Applies ratio test, root test, comparison test, and integral test for series analysis
Limit techniques
L'Hôpital, Squeeze, Algebraic
Uses L'Hôpital's rule, squeeze theorem, and algebraic manipulation for limit calculation
Test selection
Automatic Test Selection
Automatically selects the most appropriate convergence test for each sequence type
Analysis features
Behavior + Convergence
Provides detailed analysis of sequence behavior and convergence properties
Advanced analysis
Multiple Criteria
Analyzes sequences using multiple convergence criteria and mathematical principles
For sequence aₙ = 1/n (harmonic sequence):
Convergence
Diverges
Limit
0
Our sequence convergence calculator uses advanced mathematical analysis to determine sequence convergence and calculate limits. The calculation applies convergence test principles to analyze sequence behavior as the index approaches infinity.
Limit Test: lim(n→∞) aₙ = L (finite)Divergence Test: lim(n→∞) aₙ ≠ 0 → divergesRatio Test: lim(n→∞) |aₙ₊₁/aₙ| < 1 → convergesComparison Test: compare with known sequencesThis framework provides systematic approaches to determine sequence convergence using established mathematical tests and limit theory principles.
Shows how sequences approach their limits as n approaches infinity
Sequence convergence analysis is based on limit theory and mathematical analysis. The fundamental definition states that a sequence {aₙ} converges to L if for every ε > 0, there exists N such that |aₙ - L| < ε for all n > N. Various tests help determine convergence without computing the exact limit.
Need help with other mathematical analysis? Check out our infinite sum calculator and power series calculator.
Get Custom Calculator for Your PlatformResult: Sequence converges to 1
The sequence n/(n+1) approaches 1 as n approaches infinity. The +1 in the denominator becomes negligible for large n.
aₙ = 1/n
Sequence converges to 0, series diverges
aₙ = (-1)ⁿ/n
Sequence converges to 0 by alternating series test
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