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Method of substitution

The method of substitution is the among the most basic technique that can be applied to infinite power series. Its use is simple to grasp and easy to master. We will first start with a couple of simple series and build up from there.

Series representation of ex

We know that:

ex=1+x+x22!+x33!+x44!+x55!+x66!+O(x8)

ex=1+(x)+(x)22!+(x)33!+(x)44!+(x)55!+(x)66!+O(x8)

=1x+x22!x33!+x44!x55!+x66!+O(x8)

Series representation of ex2

ex=1+x+x22!+x33!+x44!+x55!+x66!+O(x8)

ex2=1+(x2)+(x2)22!+(x2)33!+(x2)44!+(x2)55!+(x2)66!+O(x8)

=1x2+x42!x63!+x84!x105!+x126!+O(x14)

Series representation of 12πex22 (Normal distribution function)

ex2=1x2+x42!x63!+x84!x105!+x126!+O(x14)

12πex22=12π(1x22+x442!x683!+x8164!x10325!+x12646!+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)=xx33!+x55!x77!+x99!x1111!+x1313!+O(x15)

sin(2x)=2x(2x)33!+(2x)55!(2x)77!+(2x)99!(2x)1111!+(2x)1313!+O(x15)

=2x8x33!+32x55!128x77!+512x99!2048x1111!+8192x1313!+O(x15)

=2x4x33+4x5158x7315+4x928358x11155925+8x136081075+O(x15)

k=0(1)(2x)2n+1(2n+1)!


Series representation of cos(2x)

cos(x)=1x22!+x44!x66!+x88!x1010!+x1212!...

cos(2x)=1(2x)22!+(2x)44!(2x)66!+(2x)88!(2x)1010!+(2x)1212!...

=14x22!+16x44!64x66!+256x88!(1024x)1010!+(4096x)1212!+...

=12x2+2x434x645+2x83154x1014175+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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