An infinite nest of square roots — Ramanujan says it's just 3
In 1911, a young clerk in Madras sent a journal this challenge: work out the value of the square root of 1, plus 2 times the square root of 1, plus 3 times the square root of 1, plus 4 times the square root of 1, plus 5... continuing forever, nested inside itself without end. No reader could solve it for six months. What does this infinite nested radical actually equal?
Reveal the answer
Exactly 3 — no decimals, no remainder. Srinivasa Ramanujan had built the puzzle (Question 289) from a far more general formula he had already derived, and when the Journal of the Indian Mathematical Society got no solutions after six months, he published the method himself. It's now one of the most quoted 'impossible-looking' identities in recreational mathematics.
— Srinivasa Ramanujan, Question 289 — Journal of the Indian Mathematical Society, vol. 3, 1911