Mathematical Tools for Physics

2—Infinite Series 32

As you see from the last two examples you have to cast the problem into a form fitting the expansion that
you know. When you want to use the binomial series, rearrange and factor your expression so that you have

(

1 +something small

)α

2.2 Deriving Taylor Series
How do you derive these series? The simplest way to get any of them is to assume that such a series exists and
then to deduce its coefficients in sequence. Take the sine for example, assume that you can write

sinx=A+Bx+Cx^2 +Dx^3 +Ex^4 +···

Evaluate this atx= 0to get

sin 0 = 0 =A+B0 +C 02 +D 03 +E 04 +···=A

so the first term,A= 0. Now differentiate the series, getting

cosx=B+ 2Cx+ 3Dx^2 + 4Ex^3 +···

Again setx= 0and all the terms on the right except the first one vanish.

cos 0 = 1 =B+ 2C0 + 3D 02 + 4E 03 +···=B

Keep repeating this process, evaluating in turn all the coefficients of the assumed series.

sinx=A+Bx+Cx^2 +Dx^3 +Ex^4 +···

cosx=B+ 2Cx+ 3Dx^2 + 4Ex^3 +···

−sinx= 2C+ 6Dx+ 12Ex^2 +···

−cosx= 6D+ 24Ex+ 60Fx^2 +···

sinx= 24E+ 120Fx+···

cosx= 120F+···

sin 0 = 0 =A

cos 0 = 1 =B

−sin 0 = 0 = 2C

−cos 0 =−1 = 6D

sin 0 = 0 = 24E

cos 0 = 1 = 120F