a. Draw illustrations like those in Figures 10.11a and 10.11b to show that the partial sums of the harmonic series satisfy the inequalities

b. There is absolutely no empirical evidence for the divergence of the harmonic series even though we know it diverges. The partial sums just grow too slowly. To see what we mean, suppose you had started with s₁ = 1 the day the universe was formed, 13 billion years ago, and added a new term every second. About how large would the partial sum sn be today, assuming a 365-day year?

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In (n + 1) en+1 1 = f" / dx = 1 + 1/ . 1 1/1/2+.. =1+ [1α=1+1₂ xdx = 1 + ln n. + n