Gujarat Power Engineeing And

Research InstituteSub :- Calculus

Topic :- Taylor’s and Maclaurin’s Series

Nikhil Patel (161040119038) Parth Patel (161040119040)Smit Patel (161040119043)Amit Prajapati (161040119047)

Created By :-Guided By :- Prof. Janki Gajjar

Details

1) Taylor’s Series I. Definition II. Examples

2) Maclaurin’s Series I. Definition II. Examples

3) Standard Expand of Maclarin’s Series

Taylor’s Series• Definition :- Let f be a function such that it

is infinitely many time differentiable in some open interval I at some internal point x=a. then the Taylor’s series generated by f at x=a is,

∑𝑛=0

∞ 𝑓 (𝑛 )(𝑎)𝑛 ! (𝑥−𝑎)𝑛= f (a )+ f ′ (a ) ( x −a )+f ′ ′ (a ) (𝑥−𝑎 )2

2 !+…

• Let, x-a=h

EXAMPLES(1) Expand in powers of x-3.

Ans. Hear x-a=x-3 So,a=3

We have to fine series about the point a=3.

We put this value In Taylor’s series, We get

(2) Using Taylor’s theorem find value of .

Ans.

(3) Expand in power of .Hence find the value of . Ans: ) let

Maclaurin’s Series• Definition :- Let f be a function such that it

has derivatives of all orders at x=0. then the Maclaurin’s series generated by f is,

EXAMPLES(1) Find a Maclaurin’s series of .

We put this value in Maclaurin’s series we get,

(2) Obtain a Maclaurin’s series for .

Ans. Here,

We put this value in Maclaurin’s series we get

(3) Find Maclaurins’s series of .

Ans. Here,

We put this value in Maclaurin’s series we get,

Standard Expanssion of Maclaurin’s series

•

Thank you...