Question

Prove that the harmonic scrics \(\sum_{n=1}^{\infty} 1 / n\) is divergent by comparing it with the series $$1+\frac{1}{2}+\left(\frac{1}{4}+\frac{1}{4}\right)+\left(\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}\right)+\left(8 \text { terms each equal to } \frac{1}{16}\right)+\cdots$$ $$\text { which is } 1+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\dots$$

Short answer

The harmonic series is divergent because its comparative summation series is divergent as well.

Step-by-step solution

Understand the Harmonic Series

The harmonic series is defined as \[ \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + \frac{1}{11} + \frac{1}{12} + \frac{1}{13} + \frac{1}{14} + \frac{1}{15} + \frac{1}{16} + \frac{1}{17} + \frac{1}{18} + \frac{1}{19} + \frac{1}{20} + \frac{1}{21} + \frac{1}{22} + \frac{1}{23} + \frac{1}{24} + \frac{1}{25} onumber \]This series is infinite, meaning it continues indefinitely.

Define the Comparison Series

The comparison series given is structured as follows: \[1 + \frac{1}{2} + \bigg(\frac{1}{4} + \frac{1}{4}\bigg) + \bigg(\frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}\bigg) + \bigg(\frac{1}{16} + \frac{1}{16}+ \frac{1}{16} + \frac{1}{16} + \frac{1}{16} + \frac{1}{16} + \frac{1}{16} + \frac{1}{16}\bigg) \]The given series simplifies to: \ 1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + ...

Recognize that Each Grouping Yields a Significant Sum

Each term grouping in the comparison series equals \( \frac{1}{2} \): \((\frac{1}{4} + \frac{1}{4}) = \frac{1}{2} \) \((\frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}) = \frac{1}{2} \) followed by \( 8 \) terms that each equal \( \frac{1}{16} \) all equate to \( \frac{1}{2} \), and so on.

Compare the Two Series

We observe that the comparison series is structured so that every grouping sums to \( \frac{1}{2} \). Since the comparison series is created from the harmonic series, if the harmonic series were to converge, the generated series should also converge. However, \[1 + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + ... = \text{diverges to } \rightarrow \text {infinity.}\]

Conclude Divergence

Given that the comparison series is clearly divergent, and it represents a lower summation bound of the harmonic series, it can be concluded that the harmonic series \[ \text{sum_{n=1}^{sum_{n=1}^{sum_{n=1}^{\frac{1}{n}}\] remains divergent as well.

Key concepts

comparison test

When dealing with infinite series, proving convergence or divergence can be challenging. One useful method is the Comparison Test. This test involves comparing your series to a second series whose convergence properties are already known. If you have a series \(\text{a}_n\) and you want to establish its convergence or divergence, you find another series \(\text{b}_n\) to compare it to.

If \(\text{a}_n\) and \(\text{b}_n\) both have non-negative terms and \(\text{a}_n \leq \text{b}_n\) for all \(n\), then:

• If \(\text{b}_n\) converges, then \(\text{a}_n\) also converges.
• If \(\text{a}_n\) diverges, then \(\text{b}_n\) also diverges.

In this exercise, the harmonic series is compared with another series that is easier to understand. By proving the simpler series diverges, we show the harmonic series also diverges through the Comparison Test.

infinite series

An infinite series is simply the sum of an infinite sequence of terms. For instance, when we add numbers indefinitely, that sum is called an infinite series. It can be represented as:
\[ \text{S} = \text{a}_1 + \text{a}_2 + \text{a}_3 + \text{a}_4 + \text{a}_5 + \text{a}_6 + \text{a}_7 + \text{a}_8 + \text{a}_9 + \text{a}_{10} + \text{a}_{11} + \text{a}_{12} + \text{a}_{13} + \text{a}_{14} + \text{a}_{15} + \text{a}_{16} + \text{a}_{17} + \text{a}_{18} + \text{a}_{19} + \text{a}_{20} \]
Sometimes, an infinite series can converge to a finite sum. For example, the geometric series \( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ... \) converges to 2. However, other times it may diverge, meaning its sum increases without bound, a behavior we see in the harmonic series. Understanding the nature of the series is crucial in determining whether it converges or diverges.

convergence and divergence

Convergence and divergence are fundamental concepts in the study of series. If an infinite series converges, it adds up to a finite number. For example, the series:

• \[ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64} + ... \]

converges to 1.

In contrast, a series diverges if its sum grows without bound or does not approach a finite limit. The harmonic series:

• \[ 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + ... \]

diverges because as you add more terms, the total sum keeps increasing indefinitely.

This difference is crucial in many fields such as mathematics, engineering, and physics, where infinite series are often used.

harmonic series

The harmonic series is an example of an infinite series that diverges. It is defined as:
\[ 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + ... \]Understanding this series is important because although each term gets smaller, the total sum grows without bound.

The harmonic series is used in various areas of science and engineering. For example,

• It appears in the analysis of algorithms in computer science (like the harmonic time complexity).
• It's also observed in electrical engineering when studying waveforms and resonance.

The harmonic series demonstrates the concept that even though individual terms decrease, their collective impact can still lead to divergence.