EXPANSIONS OF FUNCTIONS

Introduction to topic :
Here we will expand successively differentiable functions in power series.
Weightage for university exam:
08 Marks
No. of lectures required to teach:
04 hrs

POWER SERIES (Defn):
An infinite series of the form
∞
∑ anxn = a0 + a1x + a2x2 + a3x3+……………+ anxn +……..
n =0
(where a0 ,a1 ,a2 ,a3,………….,an,……..are constants.)
is called a power series in x.
similarly,
∞
∑ an(x-a)n is called power series in (x-a).
n =0

MACLAURIN’S THEOREM (Defn):
If f(x) is a function having all order derivatives at x = 0, then its
Maclaurin’s series expansion is
f(x) = f (0) + x f’ (0) +x2 f”(0) +…………………..+xn fn(0)+…………
2!
n!

Some standard Maclaurin’s series expansions :
Sr
no.
01
Function
02
f(x) = sinx
03
f(x) = cosx
04
f(x) = sinhx
05
f(x) = coshx
06
f(x) = log(1+x)
07
08
f(x) = ex
1
1 x
1
f ( x) 
1 x
f ( x) 
nth derivative
ex = 1 + x +x2 +………….+xn+…………
2!
n!
Sinx = x - x3 + x5 -………..
3! 5!
Cosx = 1 - x2 + x4 +…………
2! 4!
Sinhx = x + x3 + x5 +………..
3! 5!
Coshx = 1 + x2 + x4 +…………
2! 4!
log(1+x) = x - x2 + x3 - x4 +…………
2
3
4
1
= 1- x + x2 - x3+ ………..
1 x
1
= 1+ x + x2 + x3 + ………..
1 x
( 1)
Prepared by Mr.Zalak Patel
Lecturer, Mathematics
 Problems Based On Above Formulas :
Expand following functions in ascending powers of x (Maclaurin’s series):

(1) log sec(x+ ) (2) log coshx (3) log(1 +ex)
(4) log secx
4
(5) log(1 + x +x2+x3+x4)

Problems (Problems Of Special Type):
Expand following functions in ascending powers of x:
1- x 
(1) log (1+sinx) (2) tan-1 

1 x 
(7) exsinx
(6) tanhx
(11) xcosecx
(12)
sin -1 x
1- x2
(3) sinh-1x
(4) cos-1(4x3 -3x) (5) cos3x
(8) ex sin2x (9)
(1+x)x
(10) sin(ex-1)
1
(13) e tan x
 ASSIGNMENT:
Expand following functions in ascending powers of x (Maclaurin’s series):
(1) secx
ex
(2) x
(3) ex sinhx
e 1
 2x 
(4) sin  1  x 2  (5) log (x +
-1 
(x 2  1 )

TAYLOR’S THEOREM (Defn):
If f(x) is a function having all order derivatives at x = a, then its
Taylors’s series expansion is
f(x) = f(a) + (x-a) f’ (a) +(x-a)2 f”(a) +………….+(x-a)n fn(a)+…………
2!
n!
------------(1)

Cor-I :
If we put a=0 in above in above (1) ,
we get maclaurin’s series expansions
i.e. f(x) = f (0) + x f’ (0) +x2 f”(0) +………….+xn fn(0)+…………
2!
n!

Cor-II:
If we put x - a = h => x = a + h in above (1) then we get,
f(a+h) = f(a) + hf’ (a) +h2 f”(a) +……….+hn fn(a)+…………
2!
n!
------------(2)
( 2)
Prepared by Mr.Zalak Patel
Lecturer, Mathematics


Cor-III:
If we replace a by x in above (2) then,
f(x+h) = f(x) + h f’ (x) +h2 f”(x) +……….+hn fn(x)+…………
2!
n!
------------(3)
If we interchange x & h in above (3)
f(h+x) = f(h) + xf’ (h) +x2 f”(h) +……….+xn fn(h)+…………
2!
 Problems Based On Above Formulas :
n!
Expand following functions in ascending powers of (x-a). (Taylor’s series):
(1) ex
(2) sinx
(3) cosx.

Problems Based On Above Formulas :
(1) Expand tan-1x in powers of (x- π/4).
(2) Expand sin-1x in powers of (x- π/6).
(3) Expand x3-3x2+4x+3 in powers of (x - 2).
 x2 
x2
f”(x)
 x 
 = f(x) – 
(4) Prove that: f 
f’(x)
+
+ ……….

2
2!
(1  x)
1 x 
1 x 
(5) Prove that:
tan-1(x + b) = tan-1x + (b siny) siny – (b siny)2sin2y + (bsiny)3sin3y +………
1
2
3
( 3)
Prepared by Mr.Zalak Patel
Lecturer, Mathematics