Ramanujan's nested radical

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archive posted 4 weeks ago

Find the value of

1 + 2 1 + 3 1 + 4 1 + \dots

and explain in what sense the infinite expression has a value.

Posed by Srinivasa Ramanujan, 1911

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algebra sequences

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Solved by Srinivasa Ramanujan.

Ramanujan sent it as a question to the Journal of the Indian Mathematical Society, and when no reader answered he gave the answer himself. The key is the identity $n + 1 = 1 + n (n + 2)$ , used again and again. The infinite expression means the limit of its finite truncations, and they converge to $3$ .

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Nested radical

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