Ramanujan's Mathematical Focus: Summation, Series, and Beyond

Of the options listed, summation and series were undoubtedly the areas Ramanujan worked with the most and most famously. While he made profound contributions to other areas like recursion (continued fractions are a form of recursion), his most legendary and transformative work was in the manipulation of infinite series and novel summation methods.

Here’s a breakdown of his work in these areas, with famous examples.

1. Summation and Series (The Core of His Work)

Ramanujan had an almost supernatural intuition for infinite series. His work here can be divided into several types:

a) Summation of Divergent Series

Ramanujan developed his own method for assigning a meaningful "sum" to series that diverge in the conventional sense. This method, now called the Ramanujan summation, is a precursor to modern rigorous techniques like zeta function regularization.

Famous Example: The Sum of All Natural Numbers
Perhaps his most stunning result is "summing" the divergent series of all positive integers:
1 + 2 + 3 + 4 + 5 + ... = -1/12

How? This result arises from the analytic continuation of the Riemann zeta function, where ζ(s) = Σ n⁻ˢ. When you plug in s = -1, you get ζ(-1) = -1/12. Ramanujan's own methods arrived at this same value, which is crucial in advanced physics (e.g., string theory, the Casimir effect).

b) Rapidly Converging Series for π

He discovered incredibly efficient formulas for calculating π. Ramanujan's series converge with astonishing speed, meaning each new term adds a huge number of correct digits.

Famous Example: A Ramanujan Series for π
1/π = (2√2 / 9801) × Σ [ (4k)! × (1103 + 26390k) ) / ( (k!)⁴ × 396⁴ᵏ ) ] for k = 0 to ∞.

This series is breathtakingly efficient. Just the first term (k=0) gives π correct to 6 decimal places. Each subsequent term adds roughly 8 more decimal places of accuracy.

c) Mock Theta Functions

Toward the end of his life, Ramanujan introduced a new class of functions he called "mock theta functions". These are intricate q-series which behaved almost like modular forms. Their true deep structure was only fully understood in the early 2000s and is now a major area of research.

Example: A Third-Order Mock Theta Function
f(q) = 1 + Σ [ qⁿ² / ((1+q)²(1+q²)²...(1+qⁿ)²) ] for n = 1 to ∞ (where |q| < 1).

2. Recursion (Continued Fractions)

Ramanujan was a master of continued fractions, which are inherently recursive structures. He filled his notebooks with stunning and exotic identities involving them.

Famous Example: The Rogers-Ramanujan Continued Fraction
R(q) = q^(1/5) / (1 + q/(1 + q²/(1 + q³/(1 + ...))))

He found profound identities relating this fraction to infinite products and partition functions.

3. Other: Partition Function and Number Theory

His work on the partition function p(n) is so monumental it must be mentioned. A partition of a number n is the number of ways to write it as a sum of positive integers.

Famous Example: Congruence Properties and Asymptotic Formula
Congruences: He discovered surprising patterns:
  • p(5k + 4) ≡ 0 mod 5
  • p(7k + 5) ≡ 0 mod 7
  • p(11k + 6) ≡ 0 mod 11

Asymptotic Formula: With G.H. Hardy, he derived an exact formula for p(n) using the circle method (Hardy-Ramanujan method). This method was so accurate it could compute the exact value of p(200), a number with 13 digits.

Summary

Area	Importance to Ramanujan	Key Example
Summation/Series	Primary and most famous area. Revolutionized how we think about series and their sums.	Sum of natural numbers = -1/12; incredibly fast-converging series for π.
Recursion	Major area of contribution, primarily through the theory of continued fractions.	The Rogers-Ramanujan continued fraction and its many identities.
Other (Partitions)	One of his deepest legacies. Transformed additive number theory.	Congruences like p(5k+4) ≡ 0 mod 5; the Hardy-Ramanujan asymptotic formula.

Conclusion: While he worked extensively in all these areas, summation and series were the heart of his genius. His intuitive and revolutionary work on divergent series and hyper-efficient series for fundamental constants like π are what he is most celebrated for, and they continue to influence mathematics and physics over a century later.