# SRINIVASA RAMANUJAN 2

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### SRINIVASA RAMANUJAN:

SRINIVASA RAMANUJAN THE MAN WHO KNOW INFINITY

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Ramanujan (literally, "younger brother of Rama", a Hindu deity) was born on 22 December 1887 into a Tamil Brahmin Iyengar family in Erode , Madras Presidency (now Tamil Nadu ), at the residence of his maternal grandparents. On 14 July 1909, Ramanujan married Janaki ( Janakiammal ) (21 March 1899 – 13 April 1994), a girl whom his mother had selected for him a year earlier and who was ten years old when they married. It was not unusual for marriages to be arranged with girls. She came from Rajendram , a village close to Marudur ( Karur district ) Railway Station.

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Ramanujan departed from Madras aboard the S.S. Nevasa on 17 March 1914. When he disembarked in London on 14 April, Neville was waiting for him with a car. Four days later, Neville took him to his house on Chesterton Road in Cambridge.

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In mathematics, there is a distinction between insight and formulating or working through a proof. Ramanujan proposed an abundance of formulae that could be investigated later in depth. G. H. Hardy said that Ramanujan's discoveries are unusually rich and that there is often more to them than initially meets the eye. As a byproduct of his work, new directions of research were opened up. Examples of the most interesting of these formulae include the intriguing infinite series for π , one of which is given below: 1 π = 2 2 9801 ∑ k = 0 ∞ ( 4 k ) ! ( 1103 + 26390 k ) ( k ! ) 4 396 4 k . {\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}}.} This result is based on the negative fundamental discriminant d = −4 × 58 = −232 with class number h ( d ) = 2. Further, 26390 = 5 × 7 × 13 × 58 and 16 × 9801 = 396 2 and is related to the fact that e π 58 = 396 4 − 104.000000177 … . {\textstyle e^{\pi {\sqrt {58}}}=396^{4}-104.000000177\dots .} This might be compared to Heegner numbers , which have class number 1 and yield similar formulae

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In 1918 Hardy and Ramanujan studied the partition function P ( n ) extensively. They gave a non-convergent asymptotic series that permits exact computation of the number of partitions of an integer. Hans Rademacher , in 1937, was able to refine their formula to find an exact convergent series solution to this problem. Ramanujan and Hardy's work in this area gave rise to a powerful new method for finding asymptotic formulae called the circle method . In the last year of his life, Ramanujan discovered mock theta functions . For many years these functions were a mystery, but they are now known to be the holomorphic parts of harmonic weak Maass forms .

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