Infinite Series Calculator

Author: Henrick Yau
- 5 August 2026
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Calculators

Calculate infinite series convergence, partial sums, and analyse mathematical series. This calculator helps students, mathematicians, and engineers understand series behaviour, convergence tests, and approximations for various infinite series.

Series Definition

Use 'n' for the index variable (supports +, -, *, /, ^, sin, cos, ln, exp)
n₀
p/r
For p-series (p), geometric series (r), etc.

Analysis Options

terms
For partial sum calculations
digits

Visualization Options

to
Range for plotting partial sums

Advanced Settings

ε
Threshold for convergence detection

Infinite Series Theory

An infinite series is the sum of infinitely many terms. Understanding convergence behaviour is crucial for applications in calculus, physics, engineering, and mathematical analysis.

Types of Series
  • Geometric Series: ∑ar^n = a/(1-r) for |r| < 1
  • p-Series: ∑1/n^p converges if p > 1, diverges if p ≤ 1
  • Harmonic Series: ∑1/n diverges (p = 1 case)
  • Alternating Series: ∑(-1)^n·a_n with special convergence criteria
  • Power Series: ∑a_n(x-c)^n with radius of convergence
  • Taylor/Maclaurin Series: Function representations as power series
Convergence Tests
  • Ratio Test: lim|a_{n+1}/a_n| = L. Converges if L < 1, diverges if L > 1
  • Root Test: lim|a_n|^{1/n} = L. Converges if L < 1, diverges if L > 1
  • Integral Test: Compare with improper integral of corresponding function
  • Comparison Test: Compare with known convergent/divergent series
  • Limit Comparison Test: Use limit of ratio with comparison series
  • Alternating Series Test: If terms decrease and approach 0, series converges
Important Series Values
  • Basel Problem: ∑1/n² = π²/6 ≈ 1.6449
  • Riemann Zeta: ζ(s) = ∑1/n^s for various values of s
  • Exponential: e^x = ∑x^n/n! (converges for all x)
  • Sine Series: sin(x) = ∑(-1)^n·x^{2n+1}/(2n+1)!
  • Cosine Series: cos(x) = ∑(-1)^n·x^{2n}/(2n)!
  • Natural Log: ln(1+x) = ∑(-1)^{n+1}·x^n/n for |x| < 1
Applications
  • Numerical Analysis: Approximating functions and integrals
  • Physics: Wave functions, quantum mechanics, statistical mechanics
  • Engineering: Signal processing, control systems, Fourier analysis
  • Economics: Present value calculations, growth models
  • Computer Science: Algorithm analysis, generating functions
  • Probability: Moment generating functions, characteristic functions
Convergence Behaviour
  • Absolute Convergence: ∑|a_n| converges ⟹ ∑a_n converges
  • Conditional Convergence: ∑a_n converges but ∑|a_n| diverges
  • Uniform Convergence: Important for term-by-term operations
  • Rate of Convergence: How quickly partial sums approach the limit
  • Radius of Convergence: Domain where power series converges

Disclaimer: This calculator provides numerical approximations and theoretical analysis of infinite series. Results are subject to computational precision limitations and may not reflect exact mathematical values. For rigorous proofs and detailed analysis, consult mathematical literature and verify results independently. Some series may require advanced techniques beyond the scope of this calculator.