SERINGAPATAM, or SRIRANGAPATANA  , a town of India, formerly capital of the state of Mysore, situated on an island of the same name in the Cauvery river . Pop . (1901) 8584 . The town is chiefly noted for its fortress, which figured prominently in Indian history at the close of the 18th century . This formidable stronghold of Tippoo Sultan twice sustained a siege from the British, and was finally stormed in 1799 . After its capture the island was ceded to the British, but restored to Mysore in 1881 . The island of Seringapatam is about 3 M. in length from east to west and i in breadth, and yields valuable crops of rice and sugar-cane . The fort occupies the western side, immediately overhanging the river . Seringapatam is said to have been founded in 1454 by a descendant of one of the local officers appointed by Ramanuja, the Vishnuite apostle, who named it the city of Sri Ranga or Vishnu . At the eastern or lower end of the island is the Lal Bagh or " red garden," containing the mausoleum built by Tippoo Sultan for his father Hyder Ali, in which Tippoo himself also lies . The series is then said to converge uniformly throughout this region . If, as z approaches the value z1, n increases as lz diminishes and becomes indefinitely great as I z—zi I becomes indefinitely small the series is said to be non-uniformly convergent at the point zi .

A

function represented by a series is continuous throughout any region in which the series is uniformly convergent; there cannot be discontinuity with uniform convergence; on the other hand there may be continuity and non-uniform convergence . If ul (z) +uz(z) +... is uniformly convergent we shall have fS(z)dz=fui(z)dz+fuz(z)dz+... along any path in the region of uniform convergence ;'and we shall also have- S(z)=dZ 1(z)+dzuz(z)+...if the series dzui(s)+dzuz(z) + . . . is uniformly convergent . Uniform convergence is essentially different from absolute convergence; neither implies the other (see FUNCTION) . 18 . A series of the form ao+alz+azz2+ . . ., in which ao, a1, az, .. . are independent of z, is called a power series . In the case of a power series there is a quantity R such that the series converges if 1z 1< R, and diverges if z 1>R . A circle de-scribed with the origin as centre and radius R is called the circle of convergence. s A power series may or may not converge on the circle of convergence . The circle of convergence may be of a infinite radius as in the case of the series for sin z, viz .