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Unformatted text preview: 213 2.5. INTEGRATION ALONG CURVES
(d) Show that the sum ∞
k =1 △sk diverges.
(e) Thus, if we take (for example) the piecewise linear
approximations to the curve obtained by taking the straight line
segments as above to some ﬁnite value of k and then join the last
point to (0, 0), their lengths will also diverge as the ﬁnite value
increases. Thus, there exist partitions of the curve whose total
lengths are arbitrarily large, and the curve is not rectiﬁable. Challenge problem:
5. Bolzano’s curve (continued): We continue here our study of the
curve described in Exercise 15 in § 2.4; we keep the notation of that
(a) Show that the slope of each straight piece of the graph of fk
has the form m = ±2n for some integer 0 ≤ n ≤ k. Note that
each interval over which fk is aﬃne has length 3−k .
(b) Show that if two line segments start at a common endpoint and
end on a vertical line, and their slopes are 2n and 2n+1
respectively, then the ratio of the second to the ﬁrst length is
m1 1 + 2n+1
1 + 2n Show that this quantity is non-decreasing, and that therefore it
is always at least equal to 5/3.
(c) Use this to show that the ratio of the lengths of the graphs of
fk+1 and fk are bounded below by 2 5/3 3 + 1/3 ≥ 1.19. (d) How does this show that the graph of f is non-rectiﬁeable? ...
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