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Maclaurin’s Series Summary 1. Mathematical Representation ....... ) 0 ( ! .... .......... ) 0 ( " ! 2 ) 0 ( ' ) 0 ( ) ( ) ( 2 + + + + + = n n f n x f x xf f x f where ) 0 ( ) ( n f denotes the value of the nth derivative when 0 = x . 2. Expansions of specific standard series (Available in MF15) ..... ! ......... ! 3 ! 2 1 3 2 + + + + + + = r x x x x e r x ( ) .......... )! 1 2 ( 1 . .......... ! 5 ! 3 sin 1 2 5 3 + + + + = + r x x x x x r r ( ) .......... )! 2 ( 1 . .......... ! 4 ! 2 1 cos 2 4 2 + + + = r x x x x r r ( ) .......... 1 . .......... 3 2 ) 1 ln( 1 3 2 + + + = + + r x x x x x r r , 1 1 x Note: (i) There is no Maclaurin’s series expansion for x ln because ) 0 ( ) ( n f is undefined for all Ζ n . (ii) Substitutions can be performed to realise the series expansion for certain expressions, eg ( ) ( ) ( ) ..... ! 2 ......... ! 3 2 ! 2 2 2 1 3 2 2 + + + + + + = r x x x x e r x 3. Question structures The typical question is comprised of two parts-namely a function being given and it is required of the student to produce its Maclaurin’s series expansion up to a specified term n x through repeated differentiation; in certain instances, the proving of specific landmark differential equations are needed as well. The second part of the question usually employs the expansion found in the first part to explore the expansions of other related series-this is usually achieved through differentiation/integration of the original expansion, or a combination of other well known series expansions. Alternatively, it could also be used for approximation purposes in evaluating the value of certain numeric entities. Sample questions: a. Given that , cos 3 x e y x = show that . sin 3 3 2 x e y dx dy y x = Proving DE By further differentiation of this result, or otherwise, find the Maclaurin’s series for y, up to and including the term in . 2 x First part of question

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Page 1: Maclaurin Series Summary - A LEVEL MATHEMATICSa-levelmaths.com/Summary Handouts/Maclaurin Series Summary.pdf · Maclaurin’s Series Summary 1. Mathematical ... Note: (i) There is

Maclaurin’s Series Summary

1. Mathematical Representation

.......)0(!

..............)0("!2

)0(')0()( )(2

+++++= nn

fn

xf

xxffxf

where )0()(nf denotes the value of the nth derivative when 0=x .

2. Expansions of specific standard series (Available in MF15)

.....!

.........!3!2

132

++++++=r

xxxxe

rx

( )

..........)!12(

1...........

!5!3sin

1253

++

−+−+−=

+

r

xxxxx

rr

( )

..........)!2(

1...........

!4!21cos

242

+−

+−+−=r

xxxx

rr

( )

..........1

...........32

)1ln(

132

+−

+−+−=+

+

r

xxxxx

rr

, 11 ≤≤− x

Note: (i) There is no Maclaurin’s series expansion for xln because )0()(nf

is undefined for all Ζ∈n .

(ii) Substitutions can be performed to realise the series expansion for

certain expressions, eg ( ) ( ) ( )

.....!

2.........

!3

2

!2

221

32

2 ++++++=r

xxxxe

r

x

3. Question structures

The typical question is comprised of two parts-namely a function being given and

it is required of the student to produce its Maclaurin’s series expansion up to

a specified term nx through repeated differentiation; in certain instances, the

proving of specific landmark differential equations are needed as well. The

second part of the question usually employs the expansion found in the first part

to explore the expansions of other related series-this is usually achieved through

differentiation/integration of the original expansion, or a combination of other

well known series expansions. Alternatively, it could also be used for

approximation purposes in evaluating the value of certain numeric entities.

Sample questions:

a. Given that ,cos3 xey x= show that .sin3 32 xeydx

dyy x−=− Proving DE

By further differentiation of this result, or otherwise, find the Maclaurin’s series for

y, up to and including the term in .2x First part of question

Page 2: Maclaurin Series Summary - A LEVEL MATHEMATICSa-levelmaths.com/Summary Handouts/Maclaurin Series Summary.pdf · Maclaurin’s Series Summary 1. Mathematical ... Note: (i) There is

Hence write down the Maclaurin’s series for 3 ,cos

xe

xup to and including the term

in .2x Second part of question (exploring expansion of other

series)

b. Determine the Maclaurin’s expansion for xx tansec − , up to and including the

term in 3x . First part of question

Show that, to this degree of approximation, xx tansec − can be expressed as

)1ln( xba ++ where a and b are constants to be determined.

Second part of question(utilisation of knowledge of existing expansions)

c. Given that ,sinln 1 xy −= show that .1 2 ydx

dyx =− Proving DE

By repeated differentiation of this result or otherwise, find the Maclaurin’s

expansion for y up to and including the term in .3x First part of

question

Deduce the approximate value of .6

π

e

Second part of question ( approximating a value)

4. Small angle approximations

When x is small,

xx ≈sin , 2

1cos2x

x −≈ , xx ≈tan