ALGEBRASequencesMathematics Calculator
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Sum of Series

A series is the sum of sequence terms. Arithmetic, geometric, power, telescoping, harmonic, and p-series each have distinct convergence behavior and closed-form formulas.

Concept Fundamentals
S_n = n/2[2a+(n-1)d]
Arithmetic
S_n = a(1-r^n)/(1-r)
Geometric
Σ1/k² → π²/6
Basel
→ ln(2)
Alternating

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Euler proved Σ1/k² = π²/6 in 1735—the Basel problem. The alternating harmonic series converges to ln(2). Telescoping series get their name from terms collapsing like a telescope.

Key quantities
S_n = n/2[2a+(n-1)d]
Arithmetic
Key relation
S_n = a(1-r^n)/(1-r)
Geometric
Key relation
Σ1/k² → π²/6
Basel
Key relation
→ ln(2)
Alternating
Key relation

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Why: Series appear in finance, physics (Fourier), computer science, and signal processing.

How: Identify the series type and apply the appropriate closed-form formula.

Euler proved Σ1/k² = π²/6 in 1735—the Basel problem.The alternating harmonic series converges to ln(2).
Sources:Wolfram MathWorldKhan Academy

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Infinite Sums & ConvergenceFrom Gauss to Euler
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Sum of Series — Infinite Sums & Convergence

Arithmetic, geometric, power, telescoping, harmonic, alternating, and p-series. Explore convergence and closed-form formulas.

📐 Examples — Click to Load

Series Type

Parameters

Please enter a positive integer for number of terms

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

π²

The Basel problem: Σ₁∞ 1/k² = π²/6, solved by Euler in 1735.

ln(2)

The alternating harmonic series 1-1/2+1/3-... converges to ln(2).

📋 Key Takeaways

  • Convergence tests: Geometric converges when |r| < 1; p-series converges when p > 1
  • Closed-form formulas: Arithmetic S = n/2(2a+(n-1)d); Geometric S = a(1-r^n)/(1-r)
  • Comparison test: If 0 ≤ a_n ≤ b_n and Σb_n converges, then Σa_n converges
  • Basel problem: Σ1/k² = π²/6 — solved by Euler in 1735

💡 Did You Know?

π²The Basel problem: Σ₁∞ 1/k² = π²/6. Euler proved this in 1735.Source: Euler, 1735
📐Euler's contributions include the Euler-Maclaurin formula and Euler's constant γ.Source: Math History
ζThe Riemann zeta function ζ(s) = Σ1/k^s generalizes the Basel problem.Source: Riemann
⚛️Series appear in physics: Fourier series for waves, Taylor series for approximations.Source: Physics
💻Computer science uses series for algorithm analysis (e.g., harmonic sum in quicksort).Source: CS Theory
🎵Music theory: Fourier series decompose sounds into sine waves (harmonics).Source: Acoustics

📖 How Series Work

Arithmetic: Terms differ by constant d. Sum = n/2[2a+(n-1)d].

Geometric: Each term = previous × r. Converges when |r| < 1 to a/(1-r).

Power: Σi^p. Closed formulas for p=1,2,3.

Telescoping: Σ1/(k(k+1)) = 1 - 1/(n+1) → 1 as n→∞.

Harmonic: Σ1/k diverges despite terms → 0.

Alternating: 1 - 1/2 + 1/3 - ... converges to ln(2).

p-Series: Σ1/k^p converges iff p > 1.

🎯 Expert Tips

Identify the pattern

Check if terms have constant difference (arithmetic) or constant ratio (geometric) first.

Use closed forms

Arithmetic and geometric sums have exact formulas — avoid summing term-by-term for large n.

Convergence ≠ terms → 0

Terms must approach 0 for convergence, but that alone is not enough (harmonic diverges).

Partial fractions for telescoping

1/(k(k+1)) = 1/k - 1/(k+1) — most terms cancel.

⚖️ Series Type Comparison

TypeClosed FormulaInfinite Converges?
Arithmeticn/2[2a+(n-1)d]No
Geometrica(1-r^n)/(1-r)Yes if |r|&lt;1
Power (p=2)n(n+1)(2n+1)/6No
Telescoping1 - 1/(n+1)Yes → 1
HarmonicNoneNo
AlternatingNoneYes → ln(2)
p-SeriesNoneYes if p&gt;1

❓ FAQ

What is the difference between a sequence and a series?

A sequence is an ordered list. A series is the sum of those terms.

When does an infinite geometric series converge?

When |r| < 1. The sum is then a/(1-r).

Why does the harmonic series diverge?

Even though 1/n → 0, the sum grows without bound (like ln n).

What is the Basel problem?

Euler proved Σ₁∞ 1/k² = π²/6 in 1735.

What is a telescoping series?

A series where terms cancel in pairs, leaving only first and last terms.

When does a p-series converge?

Σ1/k^p converges if and only if p > 1.

What is the alternating harmonic series?

1 - 1/2 + 1/3 - 1/4 + ... converges to ln(2) ≈ 0.693.

How are series used in real applications?

Finance, physics (Fourier series), computer science, signal processing.

What is the sum of 1+2+3+...+100?

Using n(n+1)/2 with n=100: 100×101/2 = 5050.

📊 Series by the Numbers

π²/6
Basel Problem
ln(2)
Alternating Harmonic
5050
1+2+...+100
∞
Harmonic Diverges

⚠️ Disclaimer: This calculator provides educational estimates. For infinite series, convergence is indicated where applicable. Results may have floating-point rounding for very large n.

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