Understanding Engineering Mathematics

A Maclaurin’s series is thus just a Taylor series about the origin. Note that by changing
the variable,X=x−awe can always convert a Taylor series to a Maclaurin’s series, so
we will confine our attention to the latter here.
The coefficients in a Maclaurin’s series can be found by a nice argument, which also
illustrates the condition that a Maclaurin’s series for a functionf(x)only exists iff(x)
is differentiable an infinite number of times.
We start with the series:

f(x)=a 0 +a 1 x+a 2 x^2 +···

The job is to find the coefficientsar.a 0 is easy, just putx=0:

a 0 =f( 0 )

We now get thea 1 ,a 2 ...by differentiating enough times to isolate each of them as the
constant term, and then putx=0. Thus:

f′(x)=a 1 + 2 a 2 x+ 3 a 3 x^2 +···

so
f′( 0 )=a 1
f′′(x)= 2 a 2 + 3 × 2 a 3 x+···

so
f′′( 0 )= 2 a 2

and
a 2 =^12 f′′( 0 )
f′′′(x)= 3 × 2 a 3 +···

so
f′′′( 0 )= 3 × 2 a 3

and

a 3 =

1

3 × 2

f′′′( 0 )=

1

3!

f′′′( 0 )

You should now be able to see that in general

ar=

1

r!

f(r)( 0 )

wheref(r)( 0 )denotes therth derivative off(x)atx=0. So the Maclaurin’s series for
f(x)may be written as:

f(x)=f( 0 )+f′( 0 )x+

f′′( 0 )

2!

x^2 +···+

f(r)( 0 )

r!

xr+···

=

∑∞

r= 0

f(r)( 0 )xr

r!

wheref(^0 )( 0 )meansf( 0 ).