Calculus: Analytic Geometry and Calculus, with Vectors

12.4 Power series 627

For the case in which Fo = Fl = 1 as in (1), the formulas (3) and (6) and (9)
show that

(10) F. = (Bn+l + (-1)n,1n+iI.

Show that, when n > 1,

F. B,+1 + (-1)nAn+' B + (-1)n1gn+1/Bn

(11) 7n-i Bn - (-1)n14n - 1 - (-1)n.4n;Bn

and hence that

(12) n Fttn1 = B =^2 +

Remark: The function g in (3) is called the generating function of the sequence
Fo, Fl, F2,. The things which we have done are of interest in many branches
of mathematics and can be extended in many ways.
17 Suggest a few ways in which the expansion

sine x =! - 24i2x2 3 4 7 8

I- 6!

6 6

• 281 } ...

can be obtained. Remark: Perhaps the simplest way involves the identities

sine x = cos 2x = - (1

(2x)2 + (2x); (2x)8+ .. .l.

2! 4! 6!

18 Supposing that x > 1, obtain the formula

tan-' x=2^1

x+3x3 5xb

7x7+- ...

in two different ways. First, use the formula tan-' x = 7r/2 - tan-' (1/x) and
then use a modification of part (d) of Problem 2. Secondly, start with the
identity
((

ll2 3

+t2= t 1+ =t[1 _ t+ G) +...1

P

JJ

and integrate over the interval t? x.

19 We often use the fact that the elementary expression for the left member

of the formula

f

xu'du=logx+(s+1)(logx)2+(s+1)2(logx)3

(1) l 1! 2! 3!

+(s + 1)3(log x)' +

4!

in which x > 0, has one form when s 76 -1 and has another form when s

Prove that (1) is correct for each s. Hint: For the case in which s 0 -1,

evaluate the integral and expand the result into a series with the aid of the fact

that x-+' = e('+1) log z. Treat the case in which s = -1 separately.