C.K.PITHAWALA COLLEGE OF ENGINEERING & TECHNOLOGY, SURAT

Branch:- computer 1st Year (Div. D)ALA Subject:- Calculus

ALA Topic Name:- Power series, Taylor’s & Maclaurin’s seriesGroup No:- D9

Student Roll No Enrolment No Name

403 160090107051 Sharma Shubham 421 160090107028 Naik Rohan 455 160090107027 Modi Yash 456 160090107054 Solanki Divyesh

Submitted To

Gautam Hathiwala

Power Series Taylor’s and Maclaurin’s Series

Introduction to Taylor’s series & Maclaurin’s series

› A Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function’s derivatives at a single point.

› The concept of Taylor series was discovered by the Scottish mathematician James Gregory and formally introduced by the English mathematician Brook Taylor in 1715.

› A Maclaurin series is a Taylor series expansion of a function about zero.

› It is named after Scottish mathematician Colin Maclaurin, who made extensive use of this special case of Taylor series.

Statement of Taylor’s series

If is a given function of h which can be expanded into a convergent series of positive ascending integral power of h then,

Proof of Taylor’s series

› Let be a function of h which can be expanded into a convergent series of positive ascending integral powers of h then

𝑓 (𝑥+h )=𝑎𝑜+𝑎1 h+𝑎2 h2+𝑎3 h3+ ..........

Differentiating w.r.t. h successively,

(1)

and so on.

(2)

(3)

Putting h=0 in Eq. (1) (2) & (3),

and so onSubstituting in Eq.(1) we get,

This is known as Taylor’s series.

Putting and in the series, we get Taylor’s series in the powers of as,

NOTE : To express a function in ascending power of , express h in terms of .𝑥 𝑥

Statement of Maclaurin’s series

If is a given function of which can be expanded into a convergent series of positive ascending integral power of then,

Proof of Maclaurin series

› Let be a function of which can be expanded into positive ascending integral powers of then

𝑓 (𝑥 )=𝑎𝑜+𝑎1𝑥+𝑎2𝑥2+𝑎3𝑥3+......... .

Differentiating w.r.t. successively,

(1)

and so on.

(2)

(3)

Putting =0 in Eq. (1) (2) & (3),

and so on

Substituting in Eq.(1) we get,

This is known as Maclaurin’s series.

› The Taylor’s series and Maclaurin’s series gives the expansion of a function as a power series under the assumption of possibility of expansion of

› Such an investigation will not give any information regarding the range of values for which the expansion is valid.

› In order to find the range of values of , it is necessary to examine the behaviour of , where is the Remainder after n terms.

We have,

Where is the remainder after n terms defined as,

(1)

when this expansion (1) converges over a certain range of value of that is

then the expansion is called Taylor series of expanded about a with the range values of for which the expansion is valid.

Examples of Taylor’s series

Example 1.

Prove that:

Solution

By Taylor’s series,

Putting

Example 2:

Prove that:

Solution:

By Taylor’s series,

Putting

Hence proved.

Example 3:

Express in terms of

Solution:

By Taylor’s series,

Putting ,

(1)

,

and so on.

Substituting in Eq.(1),

Example 4.

Express in ascending powers of x.

Solution:

Let,

By Taylor’s series,

putting ,

Example 5:

Find the expansion of in ascending powers of up to terms in and find approximately

the value

Solution:

Let,

By Taylor’s series,

Putting ,

(1)

=80 and so on.

Substituting in Eq.(1),

Now

=

Maclaurin series expansion of some standard functions

Examples of Maclaurin’s series

Example 1.

.

Solution :

Let

By Maclaurin’s series,

(1)

,

Substituting in Eq.(1),

OR

Using Exponential series expansion,

Example 2.

prove that,

and conversely.

Solution :

By using exponential series expansion we get,

Conversely,

Example 3.

Prove that

Solution :

Let

Integrating the Eq.(1),

Putting

Hence,

Example 4.

Expand

Solution :

Let,

Putting

Using expansion series of ,

Example 5.

Prove that

Solution :

Example 6.Expand Solution :

End of PresentationThank You