TAYLOR’S THEOREM FOR
ONE VARIABLE
PRESENTED BY:
BHUPINDER KAUR
Associate Professor Maths
G.C.G. – 11, Chandigarh
TAYLOR’S THEOREM WITH LAGRANGE’S
Statement:
If a function f (x) is defined on [a,b] and
(i) f,f’,f’’ ….. fn-1 are all continuous in [a,b]
(ii) fn (x) exists in (a,b), then there
exists atleast one real number c in
(a,b) such that :
f(b) = f (a) + (b-a) f’ (a) + (b-a)2/2! f”(a)+……
(b  a) n1 n1
(b a) n n

f (a) 
f (c )
n  1!
n!
Deduction:
 For n = 1, Taylor’s Theorem gives
f (b)  f (a)
f (b)  f (a)  (b  a) f ' (c) 
 f ' (c )
ba
Which is Lagrange’s Mean Value Theorem
(b  a) n n
f (c )
n!
Note:
is called Lagrange’s
remainder after n term and is usually
denoted by Rn= (b  a) n n
n!
f (c )
Maclaurin’s Theorem with Lagrange’s
form of remainder after n terms.
Statement: If function f (x) is defined on
[0,x] and
(i) f,f’, f” …………..fn-1 are continuous in [0,x]
(ii) fn (x) exists in (0,x
then there exist atleast one real number
 between 0 and 1 such that
x2
x n1 n1
xn n
f ( x)  f (0)  xf ' (0)  f " (0)  ..... 
f (0)  f (x)
2!
n  1!
n!
TAYLOR’S THEOREM WITH CAUCHY’S FORM OF
REMAINDER
Statement: If a function f (x) is defined on [a,b] and
(i) f,f’, f”…….. fn-1 are all continuous in [a,b]
(ii) fn (x) exists in (a,b)
then there exist atleast one real number
t (0<t<1) such that
(b  a) 2
(b  a) n 1 n 1
f (b)  f (a)  (b  a) f ' (a) 
f " (a)  .... 
f (a)
2!
n  1!
(b  a) n
(1  t ) n 1 f n (a  t (b  a))
n 1
MACLAURIN’S THEOREM WITH CAUCHY’S
FROM OF REMAINDER AFTER TERMS.
Statement: If a function f(x) is defined on [0,x] and
(i) f,f’,f”…..fn-1 are continuous in [0,x]
(ii) fn (x) exists in (0,x); then there is exist atleast
one real number t (0<t<1) such that
x2
x n1 n1
xn
f ( x)  f (0)  xf ' (0)  f "(0)  ..... 
f (0)  (1  t ) n1 f n (tx)
2!
n 1!
n 1!
NOTE:
(b  a) n (1  t ) n 1 n
f (a  t (b  a))
n  1!
is called Cuachy’s
remainder after n terms and denotd by Rn.
CASES OF FAILURE:
(A)
Taylor’s theorem fails in the following cases:
(i)
f or one of its derivatives becomes infinite for x between
a and a + h
(ii)
f or one of its derivatives becomes discontinuous
1
betweenhan and
a + h.
n 1
Rn 1 
f (a  h)does not vanish as n   
(iii)
n  1!
(B) Maclaurin’s theorem failsin the following cases:
(I)
f or one of its derivatives becomes infinite for x near 0.
(II)
f or one of its derivatives becomes discontinuous for x
near 0.
x n 1 n 1
(iii)
Rn 1 
f (x)does not vanish as n   
n  1!
Examples:
Find lagrange’s and Cauchy’s Reminder
after n terms in the expansion of 1/1+x
Solution:
1
1 x
(1) n n!
n
f ( x) 
(1  x) n 1
f ( x) 
(1) n n!
f ( x ) 
(1  x ) n 1
n
xn n
x n (1) n n!
(1) n x n
Lagrange' s Re mainder  . f (x) 
.

n 1
n!
n! (1  x )
(1  x ) n 1
x n (1   ) n 1 n
Cauchy ' s Re mainder 
. f (x)
n  1!
x n (1   ) n 1 (1) n n!
.
n  1!
(1  x) n 1
x n (1   ) n 1 (1) n n.

(1  x) n 1