Taylor and Maclaurin Series

Posted By on December 21, 2014


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Power Series
Sequences and Series

Definition of Taylor series:

f(x)=f(a)+f(a)(xa)+f′′(a)(xa)22!++f(n1)(a)(xa)n1(n1)!+Rn
Rn=f(n)(ξ)(xa)nn! where aξx, ( Lagrangue’s form )
Rn=f(n)(ξ)(xξ)n1(xa)(n1)! where aξx, ( Cauch’s form )

This result holds if f(x) has continuous derivatives of order n at last. If limn+Rn=0, the infinite series obtained is called Taylor series for f(x) about x=a. If a=0 the series is often called a Maclaurin series.

Binomial series

(a+x)n=an+nan1+n(n1)2!an2x2+n(n1)(n2)3!an3x3+=an+(n1)an1x+(n2)an2x2+(n3)an3x3+

Special cases of binomial series

(1+x)1=1x+x2x3+1<x<1
(1+x)2=12x+3x24x3+1<x<1
(1+x)3=13x+6x210x3+1<x<1
(1+x)1/2=112x+1324x2135246x3+1<x1
(1+x)1/2=1+12x124x2+13246x3+1<x1

Series for exponential and logarithmic functions

ex=1+x+x22!+x33!+
ax=1+xlna+(xlna)22!+(xlna)33!+
ln(1+x)=xx22+x33x44+1<x1
ln(1+x)=(x1x)+12(x1x)2+13(x1x)3+x12

Series for trigonometric functions

sinx=xx33!+x55!x77!+
cosx=1x22!+x44!x66!+
tanx=x+x33+2x515++22n(22n1)Bnx2n1(2n)!π2<x<π2
cotx=1xx3x34522nBnx2n1(2n)!0<x<π
secx=1+x22+5x424+61x6720++Enx2n(2n)!π2<x<π2
cscx=1x+x6+7x3360++2(22n1)Enx2n(2n)!0<x<π

Series for inverse trigonometric functions

arcsinx=x+12x33+1324x55+135246x77+1<x<1
arccosx=π2arcsinx=π2(x+12x33+1324x55+)1<x<1
arctanx=⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪xx33+x55x77+π21x+13x315x5+π21x+13x315x5+1<x<1x1x<1
arccotx=π2arctanx=⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪π2(xx33+x55+)1x13x3+15x5π+1x13x3+15x51<x<1x1x<1

Series for hyperbolic functions

sinhx=x+x33!+x55!+
coshx=1+x22!+x44!+
tanhx=xx33+2x515+(1)n122n(22n1)Bnx2n1(2n)!+|x|<π2
cothx=1x+x3x345+(1)n122nBnx2n1(2n)!+0<|x|<π
Power Series
Sequences and Series

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Posted by Akash Kurup

Founder and C.E.O, World4Engineers Educationist and Entrepreneur by passion. Orator and blogger by hobby

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