Problem 36 - Single Variable Calculus

Question

Determine whether the given series converges or diverges. If it converges, find its sum. \(\sum_{n=0}^{\infty} \frac{2 n^{2}+n+1}{3 n^{2}+2}\)

Step-by-Step Solution

Answer

The given series \(\sum_{n=0}^{\infty} \frac{2 n^{2}+n+1}{3 n^{2}+2}\) diverges by the Limit Comparison Test.

Step 1: Test for Divergence

Compute \(\lim_{n \to \infty} \frac{2n^2 + n + 1}{3n^2 + 2}\).
Divide numerator and denominator by \(n^2\):
\(\lim_{n \to \infty} \frac{2 + 1/n + 1/n^2}{3 + 2/n^2} = \frac{2}{3}\)

Step 2: Apply the Divergence Test

Since the limit of the general term is \(\frac{2}{3} \neq 0\), the series \(\sum_{n=0}^{\infty} \frac{2n^2 + n + 1}{3n^2 + 2}\) diverges by the Divergence Test.

Key Concepts

Infinite Series Convergence

When studying infinite series, one critical question we often ask is whether the series converges, which means it approaches a specific value as more and more terms are added, or diverges, meaning it fails to settle on a finite limit. Convergence is a fundamental concept in calculus and analysis, as it ensures that the series can be meaningfully summed to yield a single, finite value.

But how do we determine if an infinite series converges? This is where tests for convergence like the Limit Comparison Test come into play. The Limit Comparison Test is particularly useful when we're dealing with series whose terms are made up of complex expressions. It involves comparing the series in question to another series with known convergence or divergence behavior. If the original series behaves similarly to this comparator series and if certain conditions are met, we can then draw conclusions about the convergence or divergence of the original series. This process not only helps in understanding the behavior of series but is also crucial in many applications of mathematics, such as engineering, economics, and the physical sciences.

Convergence and Divergence of Series

Grasping the difference between the convergence and divergence of series is critical for math students. A series is effectively just a sum of an infinite sequence of terms; if these terms approach closer and closer to zero in such a way that their cumulative sum approaches a fixed number, we say the series converges. Otherwise, if the sum grows without bound or fails to approach any particular value, the series diverges.

Why is this distinction important?

Its importance lies in the fact that convergent series can be manipulated and used in further calculations, while divergent series cannot. The series' behavior often dictates the approach mathematicians and scientists must take in their analyses. For example, the geometric series is a classic case of a series that can either converge or diverge depending on the ratio between successive terms. Learning and applying tests for convergence, such as the Limit Comparison Test, Ratio Test, or Integral Test, provides powerful tools for evaluating the behavior of series in various contexts.

Application of the Limit Comparison Test

When using the Limit Comparison Test, we find a simple series whose convergence we understand well, and compare our complex series to it term by term. If the ratio of the terms of the two series approaches a limit that is positive and finite, then both series share the same fate regarding convergence or divergence.

Calculating Limits

Calculating limits is a cornerstone of calculus and involves finding the value that a function or sequence 'approaches' as the input (or index, in the case of a sequence) approaches some value. In the context of series, limits help us understand the long-term behavior of the sequence of partial sums.

When applying the Limit Comparison Test, we seek to compute the limit of the ratio of the series' term to the term of a known reference series. This is often done by simplifying the expression, such as dividing through by the highest power of the variable in the terms, to help reveal the series' behavior as the variable grows large. As seen in the provided solution, dividing each term by \(n^2\) simplifies the expression, allowing us to easily see what happens as \(n\) approaches infinity. As \(n\) grows, terms like \(1/n\) and \(1/n^2\) become negligible, often leading to a clear limit. Understanding this principle is essential not only for working with infinite series but also for analyzing functions and model behavior in a broad range of mathematical areas.

Mastering the computation of limits is therefore not only vital for determining series convergence, but it also equips students with the analytical tools necessary to tackle problems in continuous mathematics, making it an invaluable skill in advanced mathematics education.