8.7. Taylor and Maclaurin Series http://www.ck12.org
by dividing into sum of odd and even indices. So cosx+isinx=∑∞m= 0 (− 1 )m((x 2 m)^2 )m!+i∑∞m= 0 (− 1 )m(( 2 xm)^2 +m+ 1 )^1!
The Maclaurin series of cosxand sinxfollow by separately taking the real and imaginary parts.
Exercises
- Find and compare the Maclaurin series for sinxcosxand sin 2x.
- Find the Maclaurin series ofexxfor allx.
Hint: would you dividexbyex? - Find the Maclaurin series of cos 3xand sin 3xfor allxusing Euler’s formula.
- Find expressions for the series∑∞n= 0 cosnθxnand∑∞n= 0 sinnθxnfor allθand|x|<1 using Euler’s formula.
Binomial Series
We have learned the Binomial Theorem for positive integer exponents:
(a+b)n=an+nan−^1 b+n(n2!−^1 )an−^2 b^2 +...+nabn−^1 +bn(BE)
=n
k∑= 0(n
k)
an−kbkwhere the Binomial coefficients are denoted by
(n
0
)=1 and(n
k)=n(n− 1 )...(n−k+ 1 )
k! fork≥^1
AS a simple Binomial function, takea=1 andb=x. Then( 1 +x)r=∑∞k= 0 (rk)xk
Letrbe a real number andf(x) = ( 1 +x)r. Isf(x)equal to a series in the form of (BE) except that there may be an
infinite series? The answer is yes.
Theorem(Binomial Series) Letrbe a real number and|x|<1. Then
( 1 +x)r= 1 +rx+r(rk−!^1 )x^2 +r(r−^1 3!)(r−^2 )x^3
=n
k∑= 0(r
k)
xkwhere the Binomial coefficients are denoted by
(n
0
)=1 and(n
k)=n(n− 1 )...(n−k+ 1 )
k! fork≥^1
where( 0 r)=1 and(kr)=r(r−^1 )...k(!r−k+^1 )fork≥ 1
Example 1Find a power series representation of√ 1 +x.
Solution. We need to compute the Binomial coefficients forr=−^12
( 1
2
k)
=
( 1
2)(− 1
2)(− 3
2)...(− 2 k− 3
2)
k! =(− 1 )k−^11 · 3 · 5 ...( 2 k− 3 )
2 kk!
= (−^1 )k− (^1) ( 2 k− 2 )!
2. 4. 6 ...( 2 k− 2 )) 2 kk!=
(− 1 )k−^1 ( 2 k− 2 )!
2 k−^1 k!(k− 1 ) 2 kk!=
(− 1 )k−^1 ( 2 k− 2 )!
22 k−^1 k!(k− 1 )!