Question

There are 9 one-digit numbers ( 1 to 9 ), 90 two-digit numbers ( 10 to 99 ). How many three-digit, four-digit, etc., numbers are there? The first 9 terms of the harmonic series \(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{9}\) are all greater than \(\frac{1}{10}\); similarly consider the next 90 terms, and so on. Thus prove the divergence of the harmonic series by comparison with the series \(\left[\frac{1}{10}+\frac{1}{10}+\cdots\left(9 \text { terms each }=\frac{1}{10}\right)\right]+\left[90 \text { terms each }=\frac{1}{100}\right]+\cdots\) $$ =\frac{9}{10}+\frac{90}{100}+\cdots=\frac{9}{10}+\frac{9}{10}+\cdots $$ The comparison test is really the basic test from which other tests are derived. It is probably the most useful test of all for the experienced mathematician but it is often hard to think of a satisfactory \(m\) series until you have had a good deal of experience with series. Consequently, you will probably not use it as often as the next three tests.

Short answer

The harmonic series diverges because its terms are greater than or equal to those of a known divergent comparison series.

Step-by-step solution

- Identify the Terms

Identify the terms given in the problem. The harmonic series is given as: \[ H_n = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \]

- Group the Terms

Group the terms based on the number of digits. For one-digit numbers (1 to 9), for two-digit numbers (10 to 99), and so on.

- Express the Grouped Terms

Express the sum of the grouped terms. For instance, the sum of the reciprocals of one-digit numbers is \[\sum_{i=1}^{9} \frac{1}{i}\]For two-digit numbers, the harmonic sum is \[\sum_{j=10}^{99} \frac{1}{j}\],and so forth.

- Compare Harmonic Series with Comparison Series

Use the comparison test. Compare each term of the harmonic series \(1 + \frac{1}{2} + \frac{1}{3} + \cdots\) with the series \(\left[\frac{1}{10} + \frac{1}{10} + \cdots (9 \text{ terms each } = \frac{1}{10})\right] + \left[90 \text{ terms each } = \frac{1}{100}\right] + \cdots\)

- Express the Comparison Series

Express the comparison series: \[\frac{9}{10} + \frac{90}{100} + \cdots = \frac{9}{10} + \frac{9}{10} + \cdots\]

- Evaluate Divergence

Evaluate the divergence by recognizing that the comparison series diverges. Since the harmonic series terms are greater than or equal to the terms of the comparison series, and the comparison series diverges, the harmonic series must also diverge.

- Use the Comparison Test Conclusion

State the conclusion: by the comparison test, since the simplified comparison series diverges, the original harmonic series also diverges.

Key concepts

harmonic series

The harmonic series is a well-known mathematical series that looks like this: \[ H_n = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \]
It’s composed of the reciprocals of the natural numbers. While the individual terms of the harmonic series get smaller and smaller, their sum grows without bound, meaning it diverges. It’s important to get familiar with this series because it often appears in tests and practical applications in various fields like physics and computer science.

comparison test

The comparison test is a method used to determine the convergence or divergence of an infinite series. In essence, you compare the terms of your series with the terms of another series that is already known to converge or diverge. Here's a step-by-step of how it works:

• Find a second series with terms that are easy to compare with your original series.
• Ensure the terms of your series are larger or smaller than the terms of the second series.
• If the second series is known to diverge and your series' terms are larger, then your series also diverges.
• Conversely, if your series' terms are smaller, and the second series converges, then your series also converges.

This method will be particularly useful when proving why the harmonic series diverges.

series convergence

Series convergence is a fundamental concept in calculus and analysis. A series converges if the sum of its terms approaches a finite number as the number of terms goes to infinity. In formal terms, an infinite series \(\sum a_n\) converges if \(\lim_{{n \to \infty}} S_n\) exists and is finite, where \(S_n\) is the partial sum \(S_n = a_1 + a_2 + \cdots + a_n\).
Failure to reach such a limit indicates divergence. Understanding whether a series converges or diverges is crucial for solving many practical problems across different scientific disciplines.

reciprocal sums

Reciprocal sums refer to sums of the form \(\sum \frac{1}{n}\). These are simply the sums of reciprocals of natural numbers. For example, the sum \(\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \cdots\), which is essentially the harmonic series. Studying and understanding reciprocal sums give us insights into various phenomena, such as growth patterns and rates of change in natural processes, and has applications in algorithm analysis as well.