Question

Use the Direct Comparison Test to determine the convergence of the given series; state what series is used for comparison. $$\sum_{n=1}^{\infty} \frac{n^{2}+n+1}{n^{3}-5}$$

Short answer

The series diverges by comparison with the harmonic series.

Step-by-step solution

Identify the Given Series

The series given to us is \( \sum_{n=1}^{\infty} \frac{n^{2}+n+1}{n^{3}-5} \). Our task is to analyze this for convergence using the Direct Comparison Test.

Choose a Comparison Series

For the Direct Comparison Test, we must find a series that we know converges or diverges. Notice that for large \( n \), the terms of the given series \( \frac{n^{2}+n+1}{n^{3}-5} \approx \frac{n^{2}}{n^3} = \frac{1}{n} \). The harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \) is a divergent series.

Establish Direct Comparison

To use the Direct Comparison Test, compare the terms of the series with those of \( \sum_{n=1}^{\infty} \frac{1}{n} \). We want to show that \( \frac{n^{2}+n+1}{n^{3}-5} \geq \frac{1}{n} \) for sufficiently large \( n \). By simplifying, we find that: \[ n^{3} - 5 \leq n^{3} \Rightarrow \frac{n^{2}+n+1}{n^{3}-5} \geq \frac{n^{2}+n+1}{n^{3}} = \frac{1}{n} + \frac{1}{n^2} + \frac{1}{n^3} \geq \frac{1}{n} \] for large \( n \).

Apply the Direct Comparison Test

Since \( \frac{n^{2}+n+1}{n^{3}-5} \geq \frac{1}{n} \) and the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \) diverges, by the Direct Comparison Test, the series \( \sum_{n=1}^{\infty} \frac{n^{2}+n+1}{n^{3}-5} \) also diverges.

Key concepts

Series Convergence

Series convergence is a fundamental concept in calculus and analysis, examining whether a series sums to a finite value.

When you add the terms of a series, it can move towards a specific number, known as convergence.
A series that adds up to a finite number is called a convergent series.
To check for convergence, we often use tests like the Direct Comparison Test.

• For a series to converge, its terms should become very small as the series progresses.
• If the terms don’t get smaller and smaller, the series may diverge.

The series from our exercise involves analyzing whether it converges or diverges by comparing it to another well-understood series.
The main goal of such comparisons is to find a reference point to which our series behaves similarly as numbers get very large.
By proving that our series behaves like a known divergent or convergent series, we can determine its behavior.

Divergent Series

A divergent series is one that does not sum up to a finite value.

Instead, as you continue to add more terms, the total can become infinitely large, or simply not settle on a specific number at all.

• In math, being able to distinguish between convergence and divergence is crucial, especially because divergent series behave unpredictably.
• Understanding if a series is divergent helps in many fields, as it suggests that certain expansions or processes modeled by the series could grow beyond boundaries.

In the context of our exercise, the Direct Comparison Test allowed us to demonstrate that our given series is divergent.
It shows that our series is not only large in general terms but also maintains this property compared to another famous divergent series, the harmonic series.

Harmonic Series

The harmonic series is a classic case of a divergent series.
It is defined as the sum of reciprocals of positive integers: \[\sum_{n=1}^{\infty} \frac{1}{n}\] Despite each term decreasing in size, the sum of all terms up to infinity does not stabilize to a finite number.
This is an interesting property because the terms keep getting smaller, yet the series as a whole diverges.

Such counterintuitive behavior makes the harmonic series a popular test case for comparing other series:

• It helps understand and highlight what causes divergence.
• The harmonic series provides a baseline to compare against—if another series is shown to have terms that are consistently larger, it too will diverge.

In our exercise, identifying the harmonic series as the benchmark allowed us to apply the Direct Comparison Test effectively. By showing that the terms of our series were larger or comparable to those of the harmonic series, we were able to conclude its divergent nature.