Ramanujan’s Infinite Series for π
A beautiful and extremely fast convergent formula for calculating π
1. The Famous Formula
Srinivasa Ramanujan discovered an astonishing infinite series that gives a very fast approximation of π. The formula is actually for 1/π:
1/π = (2√2 / 9801) × Σ (from k = 0 to ∞)
[ (4k)! × (1103 + 26390k) ] / [ (k!)⁴ × 396⁴ᵏ ]
In words, this means that 1/π is equal to a constant
(2√2 / 9801) multiplied by an infinite sum. For each value
of k (0, 1, 2, 3, ...), we calculate one term of the series and add
them all together.
2. Understanding Each Part
- Σ (Sigma symbol) – This means “sum of”. Here it tells us to add the terms for all integers k from 0 up to infinity.
-
Factorials – The expression
(4k)!means(4k) × (4k-1) × ... × 3 × 2 × 1. The term(k!)⁴means the factorial of k raised to the 4th power. - Linear part (1103 + 26390k) – For each value of k, this gives a different number: for example 1103 when k = 0, 27493 when k = 1, and so on.
- Power in the denominator 396⁴ᵏ – As k increases, this term becomes extremely large, so each new term in the series becomes very small, which makes the series converge very quickly.
-
Constant factor (2√2 / 9801) – This number scales the whole
sum to give the exact value of
1/π.
3. Speed of Convergence
One of the most amazing features of Ramanujan’s series is its incredibly fast convergence.
- If we keep only the first term (k = 0), the formula already gives a value of π that is correct to about 8 decimal places.
- Each additional term adds roughly 8 more correct digits of π.
This is far better than simple series such as
π = 4(1 - 1/3 + 1/5 - 1/7 + ...), which converges very slowly
and needs millions of terms to reach high accuracy.
4. How to Use the Formula
- Choose how many digits of π you want.
- Compute the terms of the series for k = 0, 1, 2, …, N.
- Add these terms to get a partial sum SN.
-
Multiply by the constant:
1/π ≈ (2√2 / 9801) × SN
and then take the reciprocal to get an approximation of π.
With only a few terms and a computer or calculator that supports large numbers, we can reach very high precision.
5. Why This Formula Is Special
Ramanujan’s formula comes from his deep research into elliptic functions, modular forms and hypergeometric series. The constants 396, 1103, 26390 and 9801 are not random. They are connected to special values of modular functions.
Modern record-breaking computations of π are based on formulas that are very similar in style, often called Ramanujan-type series. This makes Ramanujan’s work a direct ancestor of today’s high-precision calculations of π.
Srinivasa Ramanujan’s infinite series for π is a beautiful example of how deep number theory and complex analysis can produce stunningly powerful formulas.
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