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Encyclopedia Britannica



SERINGAPATAM, or SRIRANGAPATANA

This article appears in Volume V24, Page 672 of the Encyclopedia Britannica.

Encyclopedia Britannica - Main :: SCY-SHA
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 zzi 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. In +
3


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