Approximations and Exponential series | Mathematics (Maths) for JEE Main & Advanced PDF Download

F. Approximations

If x < 1, the terms of the above expansion go on decreasing and if x be very small, a stage may be reached when we may neglect the terms containing higher powers of x in the expansion. Thus, if x be so small that its squares and higher powers may be neglected then (1 + x)ⁿ = 1 + nx, approximately,

This is an approximate value of (1 + x)ⁿ

Ex.10 If x is so small such that its square and higher powers may be neglected then find the approximate value of

Sol.

Ex.11 The value of cube root of 1001 upto five decimal places is

Sol.

(1001)^1/3 = (1000 + 1)^1/3 = 

= 10 {1 + 0.0003333 -0.00000011 + ......} = 10.00333

 Ex.12 The sum of  is

Sol. Comparing with

1 + nx + + ......         ⇒ nx = 1/4              ...(i)

&  or

⇒ ⇒

⇒  ...(ii) {by (i)}

putting the value of x in (i) ⇒  ⇒ n =

 sum of series = (1 + x)ⁿ = (1 -1/2)^-1/2 = (1/2)^-1/2 =

G. Exponential series

(a) e is an irrational number lying between 2.7 & 2.8. Its value correct upto 10 places of decimal is 2.7182818284.

(b) Logarithms to the base 'e' are known as the Napierian system, so named after Napier, their inventor. They are also called Natural Logarithm. 

(c) 
 where x may be any real or complex number &

(d) 
 

(e)

H. Logarithmic series 

(a) ln (1 + x) = x - where

(b) ln (1 -x) =  where

Remember : 

(i)

(ii) e^{ln x} = x

(iii) ln2 = 0.693

(iv) ln 10 = 2.303