(^598) Series
must converge to log 2, we enter the construction business. Let Qi, Q2, Qa,
be a sequence of numbers. Let u(1), u(2), , u(ni) be, in order, the first
positive term of (1) together with just as many of the following positive termsas
are necessary to obtain a cumulated sum s(ni) which exceeds Qi. Let u(n, + 1),
u(ni + 2), , u(n2) be the first negative term of (1) together with just as
many of the following negative terms as are necesary to obtain a cumulated
sum s(n2) less than Q2. Let u(n2 + 1), u(-n2 + 2),. , u(ns) be the first
unused positive term of (1) together with just as many of the following positive
terms as are necessary to obtain a cumulated sum s(n3) which exceeds Q. Let
ulna + 1), ulna + 2), .. , u(n4) be the first unused negative term of (1)
together with just as many of the following negative terms as are necessary to
obtain a cumulated sum s(n4) less than Q4, and then continue the process. Now
comes the problem. Give precise information about the series u(1) + u(2) +
u(3) + and its sequence s(1), s(2), s(3), of partial sums when, for each
nonnegative integer m,
(a) Q. = 416, (b) Q. = Q, /(cc) Q. = 10-
(d) Qm = -10m (e) Q. = (-1)m (f) Q. = (-10)-
22 Show that the first of the formulas
log2=1 + + +-17-s +1-ra+r1- I 19+
log2 =0+ +0- +0+1+0-1 +0+1+0 -If+
flog2=1 +0 {-I- --{ { O+'lr - T + '+0 +sr - Wls+
log 2=1+s +a+T+1+rr-g+31-s+1 -a +
implies the remaining ones. What, if any, new ideas appear in this problem?
23 Let 0 < X < 1. Suppose that a steel ball dropped from height Is hits
a steel plate ./h/16 seconds later and immediately (without wasting time com-
pressing and then expanding to reverse its direction) starts to rebound to height
Ah to begin a similar bounce. Suppose that the ball continues to bounce in this
way. Find the total distance D traveled by the bouncing ball. Find also the
total time T, discovering that the small bounces occur so rapidly that the ball
does not bounce forever. .4ns.:
D = h1-a T= 1+
4 1-
24 A sequence xi, x2, xs,. is called a Cauchy sequence if
(1) lim (x, - x,,) = 0,
m,n- -
that is, if to each positive number e there correspondsa number P such that
(2) jxm - xnJ < e (m,n > P).
It is easy to prove that each convergentsequence is a Cauchy sequence. Suppose
(3) lim x = L.
lu
(lu)
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