2.5. INTEGRATION ALONG CURVES213(d)Showthat the sum∞summationdisplayk=1△skdiverges.(e) Thus, if we take (for example) the piecewise linearapproximations to the curve obtained by taking the straight linesegments as above to some finite value ofkand then join the lastpoint to (0,0), their lengths will also diverge as the finite valueincreases. Thus, there exist partitions of the curve whose totallengths are arbitrarily large, and the curve is not rectifiable.Challenge problem:5.Bolzano’s curve (continued):We continue here our study of thecurve described in Exercise15in§2.4; we keep the notation of thatexercise.(a)Showthat the slope of each straight piece of the graph offkhas the formm=±
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