Problem 18 - Single Variable Calculus

Question

Show that the series diverges. \(\sum_{n=1}^{\infty} \frac{n^{2}}{2 n^{2}+1}\)

Step-by-Step Solution

Answer

By applying the Limit Comparison Test with the simpler series \(\frac{1}{2}\sum_{n=1}^{\infty} 1\), we find that \(\lim_{n \to \infty} \frac{n^{2}}{2 n^{2}+1} = \frac{1}{2}\). Since the limit is a positive, finite number, and the simpler series diverges, the original series \(\sum_{n=1}^{\infty} \frac{n^{2}}{2 n^{2}+1}\) also diverges.

Step 1: Choose a Series for Comparison

Let's compare our series to the simpler series: \(\frac{1}{2}\sum_{n=1}^{\infty} 1\). This series clearly diverges, since the sum of constant (nonzero) terms is infinite.

Step 2: Apply the Limit Comparison Test

According to the Limit Comparison Test, if the limit of the ratio of the terms of the two series is a finite positive number, both series will either converge or diverge. Let's compare the terms of the given series \(\frac{n^{2}}{2 n^{2}+1}\) with the terms of the simpler series \(1\), which we multiply by \(\frac{1}{2}\) to match the leading term of the original series. Hence, find \[\lim_{n \to \infty} \frac{\frac{n^{2}}{2 n^{2}+1}}{1}.\]

Step 3: Compute the Limit

The limit, after simplifying, is as follows: \(\lim_{n \to \infty} \frac{n^{2}}{2 n^{2}+1} = \lim_{n \to \infty} \frac{1}{2 + \frac{1}{n^{2}}}.\) As \(n\) approaches infinity, the term \(\frac{1}{n^{2}}\) tends to zero, so \[\lim_{n \to \infty} \frac{n^{2}}{2 n^{2}+1} = \frac{1}{2} \]. This limit is non-zero and finite.

Step 4: Conclude the Result

Because the limit of the ratio of the terms of the original series and the simpler series is a positive finite number, the two series will behave the same way. We know the simpler series diverges, so it follows that the original series \(\sum_{n=1}^{\infty} \frac{n^{2}}{2 n^{2}+1}\) also diverges.

Key Concepts

Divergence

When we talk about divergence in the context of infinite series, we mean that the sum of the series tends to infinity or does not settle on a fixed number. This implies that we cannot assign a finite value to the sum of the series.
The given series \[\sum_{n=1}^{\infty} \frac{n^{2}}{2n^{2}+1}\]is used to illustrate this concept. Here, for large values of \(n\), the terms still eventually add up to something infinite rather than steadily approaching a particular value.
Key takeaways about divergence include:

• All series do not converge (some may diverge).
• Divergence means the series is unbounded.
• Tests like the Limit Comparison Test are used to confirm divergence.

Infinite series

An infinite series is essentially the sum of an infinite sequence of numbers. In this context, the series:\[\sum_{n=1}^{\infty} a_n\]means that you're summing up all terms \(a_n\) from \(n=1\) to infinity. For our exercise, this is represented by the series\[\sum_{n=1}^{\infty} \frac{n^{2}}{2n^{2}+1}\].
Understanding infinite series involves grasping:

• A sequence can be converted to a series by summing it.
• Infinite series can either converge to a finite limit or diverge.
• The behavior of an infinite series can be complex and requires careful analysis.

This infinite series does not converge to a single value and instead diverges, as seen through the methods and tests applied.

Limit of a sequence

The limit of a sequence is an important concept in understanding how sequences behave as they move toward infinity. A sequence \((a_n)\) has a limit \(L\) if, for every positive number \(\epsilon\), there's a point in the sequence beyond which all terms are within \(\epsilon\) of \(L\).
In the context of our exercise, the sequence we are concerned with is \[\frac{n^{2}}{2n^{2}+1}\],whose behavior as \(n\) becomes extremely large is essential for the test applied.
Key points about limits:

• Limits describe the behavior of terms far out in the sequence.
• Calculating a limit helps you predict the behavior of a series.
• Limits are foundational for tests like the Limit Comparison Test.

Comparison test

The Comparison Test is a tool to determine the convergence or divergence of an infinite series by comparing it with another series whose behavior is already known. There are two versions:

• Direct Comparison Test
• Limit Comparison Test

In our original task, the "Limit Comparison Test" was used. This test is ideal when terms of the series are not easy to simplify or compare directly on their own terms.
For our exercise:

• We compared \(\sum_{n=1}^{\infty} \frac{n^{2}}{2n^{2}+1}\) to the simpler divergent series, \(\sum_{n=1}^{\infty} \frac{1}{2}\).
• The limit of the ratio between these terms was calculated and found to be a positive finite number.
• This indicates that both series behave similarly, confirming the divergence of the original series.

These methods streamline the process of analyzing complex series by leveraging known results and are essential for efficient mathematical analysis.