Taylor-Maclaurin Series

Taylor-Maclaurin Series
Consider a function f defined by a power series of the form
f(x)= ?_(n=0)^??c_n (x-a)^n
with radius of convergence R > 0. If we write out the expansion of f(x) as
f(x)= c_0 +c_1 (x-a) + c_2 (x-a)^2+ c_3 (x-a)^3+ c_4 (x-a)^4+?
we observe that f(a)= c_0. Moreover
f^ (x)= c_1 + 2c_2 (x-a)+3c_3 (x-a)^2+ 4c_4 (x-a)^3+?
f^ (x)= 2c_2+3?2c_3 (x-a)+4×3c_4 (x-a)^2+?
f^((3) ) (x)=3?2c_3+ 4?3?2c_4 (x-a)+?
After computing the above derivatives we observe that
f^ (a)= c_1
f^ (a)= 2c_2 ?c_2=( f^ (a))/( 2! )
f^((3) ) (a)=3×2c_3 ? c_3=( f^((3) ) (a) )/( 3! )
In general we have
c_n=( f^((n) ) (a) )/( n! )
Suppose that f(x) has a power series expansion at x = a with radius of convergence R > 0, then the series expansion of f(x) takes the form:
f(x)= ?_(n=0)^??( f^((n) ) (a) )/( n! ) (x-a)^n=?_(n=0)^??? c_n ? (x-a)^n
f(x)=f(a)+ f^ (a)(x-a) + ( f^ (a) )/( 2! ) (x-a)^2 +( f^((3) ) (a) )/( 3! ) (x-a)^3+?
Which is called Taylor series.
If a = 0 ,then
f(x)=?_(n=0)^??? c_n ? x^n=f(0)+ f^ (0)x + ( f^ (0) )/( 2! ) x^2+( f^((3) ) (0) )/( 3! ) x^3+?
Which is called Maclaurin Series.

Example 1: Compute the Maclaurin series of the following functions
1. f(x)= e^x 2. f(x)= e^(x^2 )
f(x)= e^x ? f(0)=e^0=1
f^ (x)= e^x ? f^ (0)=e^0=1
f^ (x)= e^x ? f^ (0)= e^0=1
f^((3) ) (x)= e^x ? f^((3) ) (0)= e^0=1
1. e^x=1/0!+x/1!+x^2/2!+x^3/3!+x^4/4!+?=?_(n=0)^??x^n/n!
2. e^(x^2 )=1/0!+x^2/1!+(x^2 )^2/2!+(x^2 )^3/3!+(x^2 )^4/4!+?=?_(n=0)^??x^2n/n!
Example 2: Compute the Maclaurin series of the following functions
1. f(x)= sin?x 2. f(x)= sin?(x^2 )/x^2
f(x)=sin?x ? f(0)=sin?x=0
f^ (x)= cos?x ? f^ (0)=cos?0=1
f^ (x)= -sin?x ? f^ (0)= -sin?0=0
f^((3) ) (x)=-cos?x ? f^((3) ) (0)=- cos?0=-1
we note that f^((2n+1) ) (x)=(-1)^n cos?x ? f^((2n+1) ) (0)=(-1)^n
f^((2n) ) (x)=(-1)^n sin?x ? f^((2n) ) (0)=0
1. sin?x=x/1!-x^3/3!+x^5/5!-x^7/7!+?=?_(n=0)^??((-1)^n x^(2n+1))/(2n+1)!
2. sin?(x^2 )=((x^2 ))/1!-(x^2 )^3/3!+(x^2 )^5/5!-(x^2 )^7/7!+?=?_(n=0)^??((-1)^n (x^2 )^(2n+1))/(2n+1)!
sin?(x^2 )=x^2/1!-x^6/3!+x^10/5!-x^14/7!+?=?_(n=0)^??((-1)^n x^(4n+2))/(2n+1)!
sin?(x^2 )/x^2 =1/1!-x^4/3!+x^8/5!-x^12/7!+?=?_(n=0)^??((-1)^n x^4n)/(2n+1)!

Taylor polynomials and Maclaurin polynomials.
The partial sums of Taylor (Maclaurin) series are called Taylor (Maclaurin) polynomials. More precisely, the Taylor polynomial of degree k of f(x) at x=a is
the polynomial
P_k (x)= ?_(n=0)^k?( f^((n) ) (a) )/( n! ) (x-a)^n
and the Maclaurin polynomial of degree k of f(x) (at x = 0) is the polynomial
P_k (x)= ?_(n=0)^k?( f^((n) ) (0) )/( n! ) x^n
Example 3: Compute the Maclaurin polynomial of degree 4 for the function
f(x)=cos?x ln?(1-x)
Maclaurin polynomial P_4 (x) of degree 4 of f(x) is
P_4 (x)= ?_(n=0)^4?( f^((n) ) (0) )/( n! ) x^n=f(0)/0!+(f^ (0))/1! x+(f^ (0))/2! x^2+(f^((3) ) (0))/3! x^3+(f^((4) ) (0))/4! x^4
f_1 (x)= cos?x ? f_1 (0)=1
?f_1?^((2n+1) ) (x)=(-1)^n sin?x ? ?f_1?^((2n+1) ) (0)=0
?f_1?^((2n) ) (x)=(-1)^n cos?x ? ?f_1?^((2n) ) (0)=(-1)^n
f_1 (x)= cos?x=1-x^2/2!+x^4/4!=1-x^2/2+x^4/24
f_2 (x)= ln?(1-x) ? f_2 (0)=ln?(1)=0
f_2^ (x)= (-1)/(1-x)=-(1-x)^(-1) ? f_2^ (0)=-1
f_2^ (x)=-(1-x)^(-2) ? f_2^ (0)=-1
f_2^((3) ) (x)= -2(1-x)^(-3) ? f_2^((3) ) (0)=-2
f_2 (x)= ln?(1-x)=0-x-x^2/2!-?2x?^3/3!-?6x?^4/4!=-x-x^2/2-x^3/3-x^4/4
f(x)=cos?x ln?(1-x)=(1-x^2/2+x^4/24)(-x-x^2/2-x^3/3-x^4/4)
P_4 (x)=-x-x^2/2-x^3/3-x^4/4+x^3/2+x^4/4=-x-x^2/2+x^3/6
Example3? Compute the first four terms in the power series expansion of following
f(x)=ln?(1+x)/((1+x) )
f_1 (x)= ln?(1+x) ? f_1 (0)=ln?(1)=0
f_1^ (x)= 1/((1+x) )=(1+x)^(-1) ? f_1^ (0)=1
f_1^ (x)=-(1+x)^(-2) ? f_1^ (0)=-1
f_1^((3) ) (x)= 2(1+x)^(-3) ? f_1^((3) ) (0)=2
f_1^((4) ) (x)=-6(1+x)^(-4) ? f_1^((4) ) (0)=-6
f_1 (x)= ln?(1+x)=x-x^2/2+x^3/3-x^4/4
f_2 (x)=1/((1+x) )=(1+x)^(-1) ? f_2 (0)=1
f_2^ (x)= -(1+x)^(-2) ? f_2^ (0)=-1
f_2^ (x)=2(1+x)^(-2) ? f_2^ (0)=2
f_2^((3) ) (x)= -6(1+x)^(-3) ? f_2^((3) ) (0)=-6
f_2 (x)=1/((1+x) )=1-x+x^2-x^3+x^4
f(x)=ln?(1+x)/((1+x) )=(x-x^2/2+x^3/3-x^4/4)(1-x+x^2-x^3+x^4 )
P_4 (x)=x-x^2/2+x^3/3-x^4/4-x^2+x^3/2-x^4/3+x^3-x^4/2-x^4

P_4 (x)=x-(3x^2)/2+?11x?^3/6-(25x^4)/12

Compute the Maclaurin series of the following functions
1. f(x)=1?((1-x) ) 2. f(x)=?(1+x)
Compute the first four terms in the power series expansion of following
3.f(x)=?(1+x) cos?x 4. f(x)=((sin?x ))?((1-x) )