Calculus: Analytic Geometry and Calculus, with Vectors

12.4 Power series 625

Use this idea and the known power series expansion of cosx to obtain some of
the coefficients in the expansion

sec x = 1 +x2 + As x4 -I- WVxs -1 'g xs .h

12 We can be agreeably surprised by the simplicity of the operations which
determine the first three or four of the b's in the formulas

sin x x2 x4 xs

x = 1 T!+ i -

!

+ ...

x

sin x

and yield the formula

= bo + b2x2 + b4x4 + b6x6 + ...

(^1) x 7x3 31x5 127
cscx =x+6+360+15,120+604,800x7+
which is valid when 0 < IxI < 7.
13 Start with the power series expansion of e= anduse it to obtain the formula
ezx 1=
3+Si+...
Find the formula obtained by equating the derit atives of the members of this
formula and putting x = 1 in the result. .4ns.:
1=21 {-3 4 +5 +
(^14) Determine the first six of the coefficients in the formula
1
1+x+x2 = ao+a1x+a2x2+aax3.+...
Hint: Start by writing
1 = ao + a1x + a2x2 + asx3 + a4x4 +

• asx + a1x2 + a2xa + a3x4 +


• a0x2 + a1x3 + a,.x4 +
  and obtaining the formulas in the first column
  1=ao ao=1

0=ao+al a1=-1

0=ao+a1+a2 a2=0

0=a1+a2+as a3=1

which determine the answers in the second column.

(^15) Obtain the given coefficients and two more coefficients in the formula
1 - x _- x21+x =1+3x+4x2+7xa+ ...