Problema Basileia Σ1/k²
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
About this calculator
The Basel Problem Calculator computes the sum of the infinite series Σ1/k², which converges to π²/6. This classic problem, solved by Euler in the 18th century, highlights the relationship between infinite series and fundamental mathematical constants. To use the tool, input a positive integer N representing the number of terms to sum. The approximate result will be shown with academic precision, suitable for studies or mathematical curiosity.
The formula used is Σ₁ⁿ 1/k² ≈ π²/6 - 1/(2n) + 1/(12n²) - ... (asymptotic expansion approximation). For small N, the error may be significant, but accuracy improves as N increases. The calculator is useful in calculus or mathematical analysis courses where convergent series are studied. Note that absolute convergence ensures the sum approaches π²/6 even with negative or alternating terms.
Key precautions include avoiding non-numeric inputs or excessively large values (above 10⁶), which may cause processing delays. Do not confuse this series with the harmonic series Σ1/k (divergent) or the alternating series Σ(-1)ᵏ/k² (conditionally convergent). This tool is not a replacement for advanced math software but serves as an educational aid to visualize series behavior.
Historically, the Basel Problem was pivotal in Euler's discovery of the connection between infinite series and trigonometric functions. Today, the calculator allows testing convergence hypotheses, comparing with other series, or validating manual calculations. It is recommended for educational use or exploring properties of convergent series.
Frequently asked questions
What is the Basel series used for?
It is used to study the convergence of infinite series and their relationship with the constant π.
How many terms do I need to sum for a good approximation of π²/6?
More terms improve accuracy: 1000 terms provide 2 correct decimal places.
Can I use negative numbers or fractions in the calculation?
No, the calculator requires positive integers since the original series uses k ≥ 1.
What is the difference between Σ1/k² and Σ(-1)ᵏ/k²?
The alternating series Σ(-1)ᵏ/k² conditionally converges to π²/12, while the original absolutely converges to π²/6.
Why is the convergence slow?
The series' inherent slowness requires many terms to reduce the residual error.