Although the harmonic series does not converge, the partial sums grow very, very slowly
Chapter 0, Problem 96(choose chapter or problem)
Although the harmonic series does not converge, the partial sums grow very, very slowly. Take a right-hand sum approximating the integral of f(x)=1/x on the interval [1, n], with x = 1, to show that 1 2 + 1 3 + 1 4 + + 1 n < ln n. If a computer could add a million terms of the harmonic series each second, estimate the sum after one year.
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