Question

Give an argument, similar to that given in the text for the harmonic series, to show that \(\sum_{k=1}^{\infty} \frac{1}{\sqrt{k}}\) diverges.

Short answer

Question: Prove that the series \(\sum_{k=1}^{\infty} \frac{1}{\sqrt{k}}\) diverges using the comparison test. Answer: By comparing the terms of the given series with the harmonic series, we can see that \(\frac{1}{\sqrt{k}} \geq \frac{1}{k}\) for all \(k \geq 1\). Since the harmonic series diverges, it follows that the given series must also diverge. Therefore, \(\sum_{k=1}^{\infty} \frac{1}{\sqrt{k}}\) diverges by the comparison test.

Step-by-step solution

Identify a series for comparison

We can compare the given series to the harmonic series, which is \(\sum_{k=1}^{\infty} \frac{1}{k}\). The harmonic series is known to diverge, so if we can show that the given series has greater terms than the harmonic series, we can conclude that the given series diverges as well.

Compare the terms of the given series to the harmonic series

Now, we'll compare the terms of the given series (\(\sum_{k=1}^{\infty} \frac{1}{\sqrt{k}}\)) to the terms of the harmonic series (\(\sum_{k=1}^{\infty} \frac{1}{k}\)). Let's look at the ratio of the terms in the two series: \[ \frac{\frac{1}{\sqrt{k}}}{\frac{1}{k}} = \frac{k}{\sqrt{k}} = \sqrt{k} \] Since \(\sqrt{k} \geq 1\) for all \(k \geq 1\), we can conclude that the term \(\frac{1}{\sqrt{k}}\) is greater than or equal to the term \(\frac{1}{k}\) for all \(k \geq 1\).

Apply the comparison test

Now, we can apply the comparison test. Since the given series \(\sum_{k=1}^{\infty} \frac{1}{\sqrt{k}}\) has terms greater than or equal to the harmonic series (\(\sum_{k=1}^{\infty} \frac{1}{k}\)) and the harmonic series is known to diverge, it follows that the given series must also diverge. Therefore, we have shown that \(\sum_{k=1}^{\infty} \frac{1}{\sqrt{k}}\) diverges by the comparison test.

Key concepts

Comparison Test

The Comparison Test is a handy tool when determining the convergence or divergence of an infinite series. The test compares terms of two different series to draw conclusions about their overall behavior. Here's how it works in simple steps:

• First, identify two series. One is the series you're examining, and the other is a series you already know behaves in a certain way (converges or diverges).
• Next, compare individual terms of both series. This often involves checking if the terms of the series in question are larger or smaller than those in the already known series.
• If you're trying to prove divergence, you want terms of your series to be greater or equal to terms of a known divergent series.
• Finally, conclude based on these comparisons. If the series in question has terms larger than a known divergent series, it must also diverge.

In the exercise provided, the given series \(\sum_{k=1}^{\infty} \frac{1}{\sqrt{k}}\)is compared to the harmonic series \(\sum_{k=1}^{\infty} \frac{1}{k}\). By demonstrating that each term of the given series is larger or equal to the corresponding term of the harmonic series, we invoke the Comparison Test to show that the given series diverges.

Harmonic Series

The harmonic series is one of the fundamental concepts in mathematical analysis and understanding it is crucial for working with infinite series. The harmonic series is defined as:\[\sum_{k=1}^{\infty} \frac{1}{k}\]At first glance, you might expect this series to converge since the sum of its fractions are getting smaller as \(k\) increases. However, the harmonic series is infamous for its divergence. Even though the terms become very small, they do not reduce quickly enough to result in a finite sum.Here's an easy way to grasp this:

• Consider grouping terms in the harmonic series: \(\frac{1}{2} + \frac{1}{3} + \frac{1}{4}\) and so on.
• As you group them, you'll find each group essentially adds a fixed and significant amount, never allowing the total sum to stabilize.

Understanding the behavior of the harmonic series is key for using the comparison test effectively, as in the original exercise, where it was used to determine the divergence of a different series.

Infinite Series Divergence

Divergence of an infinite series indicates that the sum of its terms grows without bounds or fails to approach a finite limit. This is an important aspect in analysis, especially when differentiating between series that converge and those that diverge. To determine divergence:

• Check individual terms: If they don't decrease in size sufficiently fast, the series often diverges.
• Apply divergence tests: There are various tests like the Comparison Test and the Integral Test, which help to conclude about divergence.
• Use known series: Compare the series in question with a well-known divergent series like the harmonic series. If terms are greater, the given series diverges too.

In contexts where understanding series is crucial, such as calculus and higher mathematics, recognizing divergence helps prevent mishaps when estimating or calculating infinite sums. The original exercise ingeniously used divergence by comparing to the harmonic series, showcasing how divergence detection helps solve complex series problems easily.