Home
Back
Leibniz Formula for π:
| From: | To: |
The Leibniz formula for π is an infinite series discovered by Gottfried Wilhelm Leibniz in the 17th century. It provides a way to approximate the mathematical constant π through an alternating series of fractions.
The calculator uses the Leibniz formula:
Explanation: The formula alternates between adding and subtracting fractions with odd denominators, multiplied by 4. The more iterations calculated, the closer the approximation gets to the true value of π.
Details: Calculating π has been a mathematical pursuit for centuries. While modern methods are more efficient, the Leibniz series demonstrates fundamental mathematical concepts and the nature of infinite series convergence.
Tips: Enter the number of iterations you want to calculate. Higher iterations will give a more accurate result but will take longer to compute. The maximum allowed is 1,000,000 iterations.
Q1: How accurate is the Leibniz formula?
A: The Leibniz series converges very slowly. It takes about 10,000 iterations to get 3-4 decimal places of accuracy.
Q2: Are there better methods to calculate π?
A: Yes, modern algorithms like the Chudnovsky algorithm can calculate π to billions of digits much more efficiently.
Q3: Why does the series alternate signs?
A: The alternating signs create a series that oscillates around the true value of π, gradually converging toward it.
Q4: What's the mathematical basis for this formula?
A: The formula comes from the Taylor series expansion of arctan(1), which equals π/4.
Q5: Can I use this for practical π calculations?
A: While mathematically interesting, this method is not practical for high-precision calculations due to its slow convergence rate.