maclaurin series summary - a level mathematicsa-levelmaths.com/summary handouts/maclaurin series...
TRANSCRIPT
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
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