Series representation of e−x
We know that:
ex=1+x+x22!+x33!+x44!+x55!+x66!+O(x8)
e−x=1+(−x)+(−x)22!+(−x)33!+(−x)44!+(−x)55!+(−x)66!+O(x8)
=1−x+x22!−x33!+x44!−x55!+x66!+O(x8)
Series representation of e−x2
ex=1+x+x22!+x33!+x44!+x55!+x66!+O(x8)
e−x2=1+(−x2)+(−x2)22!+(−x2)33!+(−x2)44!+(−x2)55!+(−x2)66!+O(x8)
=1−x2+x42!−x63!+x84!−x105!+x126!+O(x14)
Series representation of 1√2πe−x22 (Normal distribution function)
e−x2=1−x2+x42!−x63!+x84!−x105!+x126!+O(x14)
1√2πe−x22=1√2π(1−x22+x44⋅2!−x68⋅3!+x816⋅4!−x1032⋅5!+x1264⋅6!+O(x14))
Series representation of e2x
ex=1+x+x22!+x33!+x44!+x55!+x66!+O(x8)
e2x=1+2x+(2x)22!+(2x)33!+(2x)44!+(2x)55!+(2x)66!+O(x8)
=1+2x+4x22!+8x33!+16x44!+32x55!+64x66!+O(x8)
=1+2x+2x2+4x33+2x43+4x515+4x645+O(x8)
Series representation of sin(2x)
sin(x)=x−x33!+x55!−x77!+x99!−x1111!+x1313!+O(x15)
sin(2x)=2x−(2x)33!+(2x)55!−(2x)77!+(2x)99!−(2x)1111!+(2x)1313!+O(x15)
=2x−8x33!+32x55!−128x77!+512x99!−2048x1111!+8192x1313!+O(x15)
=2x−4x33+4x515−8x7315+4x92835−8x11155925+8x136081075+O(x15)
⇔∞∑k=0(−1)(2x)2n+1(2n+1)!
Series representation of cos(2x)
cos(x)=1−x22!+x44!−x66!+x88!−x1010!+x1212!...
cos(2x)=1−(2x)22!+(2x)44!−(2x)66!+(2x)88!−(2x)1010!+(2x)1212!...
=1−4x22!+16x44!−64x66!+256x88!−(1024x)1010!+(4096x)1212!+...
=1−2x2+2x43−4x645+2x8315−4x1014175+4x12467775+...
⇔∞∑k=0(−1)(2x)2n(2n)!
Series representation of tan(2x)
tan(x)=x+x33+2x515+17x7315+62x92835+1382x11155925...
tan(2x)=2x+(2x)33+2(2x)515+17(2x)7315+62(2x)92835+1382(2x)11155925...
=2x+8x33+64x515+2176x7315+31744x92835+2830336x11155925...
Series representation of cos(2x)
cos(x)=1−x22!+x44!−x66!+x88!−x1010!+x1212!...
cos(2x)=1−(2x)22!+(2x)44!−(2x)66!+(2x)88!−(2x)1010!+(2x)1212!...
=1−4x22!+16x44!−64x66!+256x88!−(1024x)1010!+(4096x)1212!+...
=1−2x2+2x43−4x645+2x8315−4x1014175+4x12467775+...
⇔∞∑k=0(−1)(2x)2n(2n)!
Series representation of tan(2x)
tan(x)=x+x33+2x515+17x7315+62x92835+1382x11155925...
tan(2x)=2x+(2x)33+2(2x)515+17(2x)7315+62(2x)92835+1382(2x)11155925...
=2x+8x33+64x515+2176x7315+31744x92835+2830336x11155925...
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