Harold S Taylor Series

Harold’s Taylor Series

Cheat Sheet

20 April 2016

Power Series
Power Series About Zero
Geometric Series if an=a / n=0+∞anxn=a0+a1x+a2x2+a3x3+a4x4+…
Power Series / n=0+∞an(x-c)n=a0+a1x-c+a2x-c2+…
Approximation Polynomial
/ fx=Pnx+Rnx
Pnx=nthdegree polynomial approximation
Rnx=± Error
NOTE: Pnx is easy to integrate and differentiate
Maclaurin Series
Maclaurin Series
Taylor Series centered about x=0 / fx≈Pn(x)=n=0+∞fn(0)n! xn
Maclaurin Series Remainder / Rnx= fn+1(x*)(n+1)! xn+1
where x≤ x*≤max and limx→+∞Rnx=0
Taylor Series
Taylor Series
Maclaurin Series if c=0 / fx≈Pn(x)=n=0+∞fn(c)n! (x-c)n
Taylor Series Remainder / Rnx=fn+1(x*)(n+1)! (x-c)n+1
where x≤ x*≤c and limx→+∞Rnx=0
Series Examples
Exponential Functions
ex=n=0∞xnn! for all x / 1+x+x22!+x33!+x44!+x55!+x66!+x77!+x88!+…
ax=ex ln⁡(a)=n=0∞(xln(a))nn! for all x / 1+x ln(a)+x ln(a)22!+x ln(a)33!+…
Natural Logarithms
ln (1-x)=n=1∞xnn for x<1 / x+x22+x33+x44+x55+x66+x77+x88+…
ln (x)=n=1∞(-1)n-1(x-1)nn for x<1 / x-1-x-122+x-133-(x-1)44+…
ln (1+x)=n=1∞(-1)n-1nxn for x<1 / x-x22+x33-x44+x55-x66+x77-x88+…
ln 1+x1-x=n=1∞22n-1x2n-1 for x<1 / 2x-2x23+2x35-2x47+2x59-2x611+2x713-…
Geometric Series
1x=n=0∞-1n(x-1)n for 0<x<2 / 1-(x-1)+(x-1)2-(x-1)3+(x-1)4+…
11+x=n=0∞(-1)nxn for x<1 / 1-x+x2-x3+x4-x5+x6-x7+x8-…
11-x=n=0∞xn for x<1 / 1+x+x2+x3+x4+x5+x6+x7+x8+…
11+x2=n=0∞(-1)nx2n for x<1 / 1-x2+x4-x6+x8-x10+x12-x14+…
11-x2=n=0∞x2n for x<1 / 1+x2+x4+x6+x8+x10+x12+x14+…
1(1+x)2=n=1∞(-1)n-1 nxn-1 for x<1 / 1-2x+3x2-4x3+5x4-6x5+7x6-…
1(1-x)2=n=1∞nxn-1 for x<1 / 1+2x+3x2+4x3+5x4+6x5+7x6+…
11+x3=n=2∞-1n-1 n-1n2xn-2
for x<1 / 1-3x+6x2-10x3+15x4-21x5+28x6-…
1(1-x)3=n=2∞(n-1)n2xn-2 for x<1 / 1+3x+6x2+10x3+15x4+21x5+28x6+…
1+x=n=0∞-1n 2n!4n (n!)2 (1-2n)xn
for-1<x≤1 / 1+12x-18x2+116x3-5128x4+7256x5-211,024x5+…
1-x=n=0∞2n!4n (n!)2 (1-2n)xn
for-1<x≤1 / 1-12x-18x2-116x3-5128x4-7256x5-211,024x5-…
1+x2=n=0∞-1n 2n!4n (n!)2 (1-2n)x2n
for-1<x≤1 / 1+12x2-18x4+116x6-5128x8+7256x10-…
1-x2=n=0∞2n!4n (n!)2 (1-2n)x2n
for-1<x≤1 / 1-12x2-18x4-116x6-5128x8-7256x10-…
Double Factorial (!!) / (n)‼=nn-2n-4…6∙4∙2 if even
(n)‼=nn-2n-4…5∙3∙1 if odd
where 0‼=1 and ─1‼=1
11+x=n=0∞-1n 2n-1‼2n‼xn
for-1<x≤1 / 1-12x+1∙32∙4x2-1∙3∙52∙4∙6x3+1∙3∙5∙72∙4∙6∙8x4-…
11-x=n=0∞ 2n-1‼2n‼xn
for-1<x≤1 / 1+12x+1∙32∙4x2+1∙3∙52∙4∙6x3+1∙3∙5∙72∙4∙6∙8x4+…
11+x2=n=0∞-1n 2n-1‼2n‼x2n
for-1<x≤1 / 1-12x2+1∙32∙4x4-1∙3∙52∙4∙6x6+1∙3∙5∙72∙4∙6∙8x8-…
11-x2=n=0∞ 2n-1‼2n‼x2n
for-1<x≤1 / 1+12x2+1∙32∙4x4+1∙3∙52∙4∙6x6+1∙3∙5∙72∙4∙6∙8x8+…
Binomial Series
(1+x)r=n=0+∞rn xn
for x<1 and all complex r where
rn=k=1nr-k+1k
=rr-1r-2…(r-n+1)n! / 1+rx+rr-12! x2+rr-1(r-2)3! x3+…
Trigonometric Functions
sin (x)=n=0∞(-1)n2n+1!x2n+1 for all x / x-x33!+x55!-x77!+x99!-x1111!+x1313!-x1515!+…
cos (x)=n=0∞(-1)n2n!x2n for all x / 1-x22!+x44!-x66!+x88!-x1010!+x1212!-x1414!+…
tan x=n=1∞(-1)n-1 22n 22n-1 B2n2n!x2n-1
for xπ2 / x+13x3+215x5+17315x7+622,835x9+1,382155,925x11+21,844608,1075x13+…
sec x=n=0∞-1n E2n2n!x2n
for xπ2 / 1+x22!+5x44!+61x66!+1,385x88!+50,521x1010!+…
csc x=n=0∞-1n-1 2 22n-1-1 B2n2n!x2n-1
for 0<x<π / 1x+16x+7360x3+3115,120x5+127604,800x7+…
cot x=n=0∞(-1)n 22n B2n2n!x2n-1
for 0<x<π / 1x-13x-145x3-2189x5-14,725x7-42,835x9-…
Inverse Trigonometric Functions
sin-1(x)=n=0∞2n!(2nn!)22n+1 x2n+1
for x≤1
sin-1(x)=n=0∞Γn+12π 2n+1 n! x2n+1 / x+x32∙3+1∙3x52∙4∙5+1∙3∙5x72∙4∙6∙7+1∙3∙5∙7x92∙4∙6∙8∙9+…
cos-1(x)=π2-sin-1(x)
for x≤1 / π2-x-x32∙3-1∙3x52∙4∙5-1∙3∙5x72∙4∙6∙7-…
tan-1(x)=n=0∞-1n2n+1x2n+1
for x<1 / x -x33+x55-x77+x99-… for-1<x<1
π2-1x+13x3-15x5+17x7-19x9+… for x≥ 1
-π2-1x+13x3-15x5+17x7-19x9+… for x<1
sec-1(x)=-i lnx+i ln2-i4n=0∞2n+1! x2n+24nn+1!2 / -i lnx+i ln2-i4x2-3i32x4-5i96x6-…
csc-1(x)=i lnx-i ln2+π2+i4n=0∞2n+1! x2n+24nn+1!2 / i lnx-i ln2+π2+i4x2+3i32x4+5i96x6+…
cot-1(x)=π2-tan-1(x)
for x<1 / π2-x+x33-x55+x77-x99+… for-1<x<1
1x-13x3+15x5-17x7+19x9-… for x≥ 1
π+1x-13x3+15x5-17x7+19x9-… for x<1
Hyperbolic Functions
sinh (x)=ex-e-x2=n=0∞x2n+12n+1! for all x / x+x33!+x55!+x77!+x99!+x1111!+x1313!+x1515!+…
cosh (x)=ex+e-x2=n=0∞x2n2n! for all x / 1+x22!+x44!+x66!+x88!+x1010!+x1212!+x1414!+…
tanh x=ex-e-xex+e-x
tanh (x)=n=1∞ 22n(22n-1) B2n2n!x2n-1
for xπ2 / x-2x33!+16x55!-272x77!+7,936x99!-353,792x1111!+…
x-13x3+215x5-17315x7+622,835x9-1,382155,925x11+…
sech (x)=n=0∞E2n2n!x2n
for xπ2 / 1-x22!+5x44!-61x66!+1,385x88!-50,521x1010!+…
csch x=1x+n=1∞2-22n B2n2n!x2n-1
for 0<xπ / 1x-16x+7360x3-3115,120x5+127604,800x7-…
coth (x)=1x+n=1∞(-1)n-1 22n B2n2n!x2n-1
for 0<xπ / 1x+13x-145x3+2945x5-14,725x7+42,835x9-…
Inverse Hyperbolic Functions
sinh-1(x)=n=0∞(-1)n2n!(2nn!)22n+1x2n+1
for x≤1
sinh-1(x)=n=0∞(-1)n 2n-1‼(2n+1)2n‼x2n+1 / x-x32∙3+1∙3x52∙4∙5-1∙3∙5x72∙4∙6∙7+1∙3∙5∙7x92∙4∙6∙8∙9-…
cosh-1(x)=π2i-in=0∞2-nn!2n+1x2n+1
for x≤1 / π i2-i x-i x32∙3-i∙1∙3x52∙4∙5-i∙1∙3∙5x72∙4∙6∙7-…
tanh-1(x)=n=0∞x2n+12n+1
for x<1, x≠±1 / x+x33+x55+x77+x99+x1111+x1313+x1515+…
sech-1(x)=- lnx+ln2+14n=0∞2n+1! x2n+24nn+1!2 / - lnx+ln2+14x2+3i32x4+5i96x6+…
csch-1(x)=- lnx+ln2+14n=0∞(-1)n 2n+1! x2n+24nn+1!2 / - lnx+ln2+14x2-3i32x4+5i96x6-…
coth-1(x)=-iπ2+n=0∞x2n+12n+1 / =-iπ2+x+x33+x55+x77+x99+x1111+x1313+…
Bernoulli Numbers / Euler Numbers / Gamma Function
B0=1
B1=-12
B2=16
B4=-130
B6=142
B8=-130
B10=566
B12=-6912,730
B14=76
B16=-3,617510
B18=438,675798
B20=-174,611330
B22=854,513138 / E0=1
E1=0
E2=-1
E3=0
E4=5
E5=0
E6=-61
E8=1,385
E10=-50,521
E12=2,702,765
E14=-199,360,981
E16=19,391,512,145
E18=-2,404,879,675,441
E20=370,371,188,237,525
E22=-69,348,874,393,137,901
E2n+1=0 / Γ0=Γ12=π
Γ1=Γ32=π2
Γ2=Γ52=3π4
Γ3=Γ72=15π8
Γ4=Γ92=105π16,
Γ5=Γ112=945π32
Γ6=Γ132=10,395π64
Γ7=Γ152=135,135π128
Γ8=Γ172=2,027,025π256
Γ9=Γ192=34,459,425π512
Γ10=Γ212=654,729,075π1,024
Generating Function
tet-1=n=0∞Bn n!tn / 1cosh⁡(t)=2et+e-t=n=0∞En n!tn / Γt=0∞xt-1 e-x dx
Γ12+n=2n!4n n!π
Γn2=n-2‼2n-12π
Recursive Definition / Iterated Sum / Recursive Definition
Bmn=nm-k=0m-1mkBknm-k+1
B0n=1 / E2n=ik=12n+1 j=0kkj (-1)j(k-2j)2n+12k ik k / Γn+1=n∙Γn
Γn2=n-22∙Γn-22
Γ12=π
References:
https://www.wolframalpha.com
https://en.wikipedia.org
http://ddmf.msr-inria.inria.fr/1.9.1/ddmf
http://web.mit.edu/kenta/www/three/taylor.html