7/28/2019 Theory Taylor Series

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Taylors Series :

f ( x ) = f ( a ) + ( x a )!1

)(' a f + ( x a ) 2 !2

)(" a f + ( x a ) 3 !3

)(''' a f

+ + ( x a ) n !

)(n

a f n

+

1 : When a = 0, the Taylors series reduces to Maclaurins series

f ( x ) = f (0) + x !1)0(' f

+ x2

!2)0(" f

+ x3

!3)('" a f

2 : There are several forms of Taylors series used in different contexts .

For example :

(i) Replace a by a ,

f ( x ) = f ( a )+( x +a )!1

)(' a f +( x +a )2!2

)(" a f +( x +a )3!3

)(''' a f +

(ii) When x =a + h then,

f (a + h) = f (a ) + h !1

)(' a f + h2 !2

)('' a f + h3 !3

)(''' a f +

(iii) Replace h by h we get,

f (a h) = f (a ) h !1

)(' a f + h2 !2

)('' a f h3 !3

)(''' a f +

(iv) Replace h by x we get ,

f (a + x ) = f (a ) + x !1

)(' a f + x 2 !2

)('' a f + x 3 !3

)(''' a f +