Calculus: Analytic Geometry and Calculus, with Vectors

596 Series

whenever the series on the right are convergent. Solution: The inequality

0 (Iukl - I I) = IZ[kI - 21,,k1 I^xI + IVkI2

implies that

2lukl IVkI < IukI2 + Iv*12

and hence

n n n

2 Inkl lokl I ukl + U + V,

k-1 k=1 k=1

where U and V are the numbers to which !IuqI2 and jIo1I2 converge. The resulc

follows.

17 Imagine that a coin is tossed repeatedly and that we let 1k = 1 if the kth

toss produces a head and let xk = 0 if the kth toss produces a tail. Tell why

the series in

must be convergent to a number x for which 0 <= x < 1. Show that

12 x1 21 41+x^ +x3 22 23 S+

and

1 2x,-1 2x2-1 2x3-1

x 2 +^22 +^23 + 24 +

Remark: This problem can steer our thoughts toward the Rademacher functions

ri(t), r2(t), for which

r12t) +T'22)+X22)+...

when 0 5 t 5 1. Figures 12.191, 12.192, and 12.193 exhibit graphs of the first

1

Y

I

0

Figure 12.191 Figure 12.192 Figure 12.193

three Rademacher functions. These things are important in the theory of

probability and elsewhere.

18 Look briefly at the following outline of a proof that e is irrational and then,

with the textbook out of sight, write a proof in which more details are given. If