Question

JEE ADVANCE :-

Write all the series expansion involved in the chapter 'limits'.​

Answer 1

The various series limit which is used in JEE ADVANCE are :-

${e}^{x} = 1 + x + \frac{ {x}^{2} }{2 ! } + \frac{ {x}^{3} }{3!} + \frac{ {x}^{4} }{4 !}.......$

$log(1 + x) = x - \frac{ {x}^{2} }{2} + \frac{ {x}^{3} }{3} - \frac{ {x}^{4} }{4} .....$

${(1 + x)}^{n} = 1 + nx + \frac{n(n - 1) {x}^{2} }{2!} + \frac{n(n - 1)(n - 2) {x}^{3} }{3!}.....$

${(1 + x)}^{ \frac{1}{x} } = e(1 - \frac{x}{2} + \frac{11}{24} {x}^{2} ....)$

$\sin \: x= x - \frac{ {x}^{2} }{2!} + \frac{ {x}^{3} }{3!} - \frac{ {x}^{4} }{4!}.....$

$\cos(x) = 1 - \frac{ {x}^{2} }{2!} + \frac{ {x}^{4} }{4!} - \frac{ {x}^{6} }{6!}$

$\tan(x) = x + \frac{ {x}^{3} }{3} + \frac{2 {x}^{5} }{15}$

${ \tan }^{ - 1} x = x - \frac{ {x}^{3} }{3} + \frac{ {x}^{5} }{5} .....$