Calculus: Analytic Geometry and Calculus, with Vectors

614 Series

when x is not an integer and [x] denotes the greatest integer less thanor

equal to x. Problem 8 at the end of this section will show where the

first of the formulas

(12.384)

BL(x) -^2 sin 27rx +

2,r 1

B2(x)

B3(x)

B4(x)

Bs(x)

2 cos 2,rx +

(2,r)2^122

2 sin 2,rx +

(2,r)3 V

-2 cos 27rx +

(2,r)4^14

-2 sin 2,rx+

(2x)6 1s

sin 4irx+sin6 ax+sink nrx+

cos 47rx cos 6rx cos 8,rx

22 + 32 + 42 +

si n4nrx+sin 36,rx+sin +

43

co 4nrx + cos 36,r x + co s8,rx +

si n4,rx+sin36nrx+sin48nrx+

comes from. The remaining formulas come by successive integration;

it can be proved that the series for B2(x), B3(x), B4(x), can be dif-

ferentiated and integrated termwise [except that the series for B2(x) is

not termwise differentiable when x is an integer] and hence that the

fundamental formulas (12.381) and (12.382) hold. Since Section 4.3,

Problem 10, shows that

(12.385) B2(0) =B'=

12' B4(0) 41 720'

putting x = 0 in the formulas for B2(x) and B4(x) gives

(12.386) (2) = I k2 = 6 , 1 3'(4) = 1ik 4 = 90

Balth Van der Pol used to claim that persons who know these formulas

are mathematicians and persons who do not are not.

Problems 12.39

(^1) Use Theorem 12.31 to show that each of the following series is convergent.
(a)
(^1111)
log 3 log 4 + log 5 log 6 +
1

1 1

(c) log log 20 log log 21 + log log 22 log log 23 +

1 (^1111)
1+x2 2+x2+3+xa 4-}-x2+5+x2

log 33 - 4log 4.+ 5 - 6 +log 5 log 6 log 7

7

• ...