Question

Harmonic sum In Chapter 10, we will encounter the harmonic sum 1 + 1/2 + 1/3 + ⋯ + 1/n. Use a left Riemann sum to approximate ∫[1 to n+1] (dx/x) (with unit spacing between the grid points) to show that 1 + 1/2 + 1/3 + ⋯ + 1/n \u003e ln(n + 1). Use this fact to conclude that lim (n → ∞) (1 + 1/2 + 1/3 + ⋯ + 1/n) does not exist.

Accepted Answer

Recall that the integral $$ \int_1^{n+1} \frac{1}{x} \, dx $$ represents the area under the curve $$ y = \frac{1}{x} $$ from $$ x=1  to$$ $$ x=n+1 . $$This integral evaluates to $$ \ln(n+1) . $$Set up a left Riemann sum with unit spacing to approximate the integral $$ \int_1^{n+1} \frac{1}{x} \, dx . $$The left endpoints for the subintervals are $$ x = 1, 2, 3, \ldots, n $$, so the sum is $$ \sum_{k=1}^n \frac{1}{k} \cdot 1 = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} . $$Since $$ \frac{1}{x}  is a $$decreasing function, the left Riemann sum overestimates the integral. Therefore, we have the inequality $$ 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \u003e \int_1^{n+1} \frac{1}{x} \, dx = \ln(n+1) . $$Use this inequality to analyze the behavior of the harmonic sum as $$ n 	o \infty . $$Because $$ \ln(n+1) 	o \infty  as$$ $$ n 	o \infty $$, the harmonic sum $$ 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} $$ must also grow without bound. Conclude that the limit $$ \lim_{n 	o \infty} \left(1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}ight) $$ does not exist as a finite number, since the harmonic sum diverges to infinity.

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주요 개념

Harmonic Series

Left Riemann Sum Approximation

Improper Integral and Limit Comparison