Ramanujan calculated pi using rapidly converging infinite series, notably one that could compute approximately eight more decimal places of pi with each additional term.
Ramanujan's Formula for Pi
Ramanujan developed several formulas for calculating pi. One such formula, which demonstrates the innovative techniques he used, is:
1/π = (2√2 / 9801) * ∑k=0∞ (4k)! / (k!)4 * (1103 + 26390k) / 3964k
Here's a breakdown of the components and why it's significant:
- ∑k=0∞: This represents an infinite sum, starting with k=0 and increasing without limit.
- (4k)! / (k!)4: This part includes factorials, where
n!means the product of all positive integers up ton. This part grows very rapidly, but the denominator helps to control the overall growth. - (1103 + 26390k): This part is a linear expression, multiplying each term of the series by an increasing coefficient.
- 3964k: This is the denominator that significantly speeds up the convergence of the series.
Significance
- Rapid Convergence: The main advantage of this formula is its rapid convergence. As noted, approximately eight new decimal places of pi are computed for each new term added in the series. This means a very accurate estimate of π can be achieved using only a few terms.
- Infinite Series: Ramanujan's use of an infinite series is a powerful technique, allowing for very high precision approximations of pi.
Example
To illustrate, let's look at the first few terms of this series:
- k=0 : (1103) / 1
- k=1 : (4!/(1!^4) * (1103 + 26390) / (396^4)
Even these first two terms provide a very good approximation of pi due to how quickly the terms become small.
Practical Application
This type of series is crucial in computational mathematics and computer science when very high precision calculations are needed.
Summary
Ramanujan calculated pi using a series of rapidly converging infinite series like the one described above. This specific series yields roughly eight new decimal places per term, making it a highly effective method for calculating very accurate estimates of pi.
