# Taylor and Maclaurin Series

Posted By on December 21, 2014

### Definition of Taylor series:

 f(x)=f(a)+f′(a)(x−a)+f′′(a)(x−a)22!+⋯+f(n−1)(a)(x−a)n−1(n−1)!+Rn
 Rn=f(n)(ξ)(x−a)nn! where a≤ξ≤x, ( Lagrangue’s form )
 Rn=f(n)(ξ)(x−ξ)n−1(x−a)(n−1)! where a≤ξ≤x, ( 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+nan−1+n(n−1)2!an−2x2+n(n−1)(n−2)3!an−3x3+⋯=an+(n1)an−1x+(n2)an−2x2+(n3)an−3x3+⋯

### Special cases of binomial series

 (1+x)−1=1−x+x2−x3+⋯−1
 (1+x)−2=1−2x+3x2−4x3+⋯−1
 (1+x)−3=1−3x+6x2−10x3+⋯−1
 (1+x)−1/2=1−12x+1⋅32⋅4x2−1⋅3⋅52⋅4⋅6x3+⋯−1
 (1+x)1/2=1+12x−12⋅4x2+1⋅32⋅4⋅6x3+⋯−1

### Series for exponential and logarithmic functions

 ex=1+x+x22!+x33!+⋯
 ax=1+xlna+(xlna)22!+(xlna)33!+⋯
 ln(1+x)=x−x22+x33−x44+⋯−1
 ln(1+x)=(x−1x)+12(x−1x)2+13(x−1x)3+⋯x≥12

### Series for trigonometric functions

 sinx=x−x33!+x55!−x77!+⋯
 cosx=1−x22!+x44!−x66!+⋯
 tanx=x+x33+2x515+⋯+22n(22n−1)Bnx2n−1(2n)!−π2
 cotx=1x−x3−x345−⋯−22nBnx2n−1(2n)!0
 secx=1+x22+5x424+61x6720+⋯+Enx2n(2n)!−π2
 cscx=1x+x6+7x3360+⋯+2(22n−1)Enx2n(2n)!0

### Series for inverse trigonometric functions

 arcsinx=x+12x33+1⋅32⋅4x55+1⋅3⋅52⋅4⋅6x77+⋯−1
 arccosx=π2−arcsinx=π2−(x+12x33+1⋅32⋅4x55+⋯)−1
 arctanx=⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪x−x33+x55−x77+⋯π2−1x+13x3−15x5+⋯−π2−1x+13x3−15x5+⋯−1
 arccotx=π2−arctanx=⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪π2−(x−x33+x55+⋯)1x−13x3+15x5−⋯π+1x−13x3+15x5−⋯−1

### Series for hyperbolic functions

 sinhx=x+x33!+x55!+⋯
 coshx=1+x22!+x44!+⋯
 tanhx=x−x33+2x515+⋯(−1)n−122n(22n−1)Bnx2n−1(2n)!+⋯|x|<π2
 cothx=1x+x3−x345+⋯(−1)n−122nBnx2n−1(2n)!+⋯0<|x|<π

#### Posted by Akash Kurup

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

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