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Chapter 9: Problem 1

Explain why \(\sum_{n=1}^{\infty} \frac{n^{2}}{2 n^{2}+1}\) diverges.

Short Answer

Expert verified
The series diverges because its terms do not approach 0.

Step by step solution

01

Identify Test for Divergence

To determine whether the series \(\sum_{n=1}^{\infty} \frac{n^{2}}{2n^{2}+1}\) diverges, consider using the Limit Comparison Test or the Divergence Test, since it involves terms that approach a constant ratio.
02

Apply Divergence Test

Recall the Divergence Test, which states that if \(\lim_{n\to\infty} a_n eq 0\), then \(\sum_{n=1}^{\infty} a_n\) diverges. For this series, set \(a_n = \frac{n^{2}}{2n^{2}+1}\) and find the limit as \(n\) approaches infinity.
03

Compute the Limit

Simplify the expression for \(a_n\) as \(n\) becomes very large. We have:\[\lim_{n\to\infty} \frac{n^{2}}{2n^{2}+1} = \lim_{n\to\infty} \frac{1}{2 + \frac{1}{n^{2}}}\]As \(n\) approaches infinity, \(\frac{1}{n^{2}}\) approaches 0. Thus, the expression simplifies to \(\frac{1}{2}\).
04

Conclude by Divergence Test

Since \(\lim_{n\to\infty} a_n = \frac{1}{2} eq 0\), the series \(\sum_{n=1}^{\infty} \frac{n^{2}}{2n^{2}+1}\) does not converge to zero, indicating divergence by the Divergence Test.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Limit Comparison Test
The Limit Comparison Test is a powerful tool for determining the convergence or divergence of an infinite series. It is particularly useful when you suspect that a complex series behaves similarly to a simpler comparison series. Suppose you have two series with positive terms, \( \sum_{n=1}^\infty a_n \) and \( \sum_{n=1}^\infty b_n \), and you're interested in whether both series converge or diverge. The test involves calculating the limit of the ratio of their terms:\[ L = \lim_{n \to \infty} \frac{a_n}{b_n} \]
  • If \( L \) is a positive finite number, both series behave the same way鈥攅ither both converge or both diverge.
  • If \( L = 0 \) and \( \sum b_n \) converges, then \( \sum a_n \) converges.
  • If \( L = \infty \) and \( \sum b_n \) diverges, then \( \sum a_n \) diverges.
This test is especially handy when comparison directly through the terms is difficult, making it easier to analyze series that might otherwise be unwieldy.
convergence of series
When discussing the convergence of a series, we are often determining whether the sum of an infinite sequence of terms approaches a finite limit. The key question is whether the series, as we sum more terms, reaches a specific value. For a series \( \sum_{n=1}^\infty a_n \), it converges if the partial sums \( S_n = a_1 + a_2 + \cdots + a_n \) approach a finite number as \( n \) approaches infinity.
  • A series converges absolutely if \( \sum_{n=1}^\infty |a_n| \) converges.
  • A series converges conditionally if \( \sum_{n=1}^\infty a_n \) converges, but \( \sum_{n=1}^\infty |a_n| \) does not.
To check for convergence, we utilize various tests such as the Divergence Test, the Ratio Test, and others. In many cases, a simple divergence test might suffice where the limit of the terms \( a_n \), as \( n \to \infty \), is non-zero indicating divergence.
infinite series
An infinite series is the sum of infinitely many terms, expressed symbolically as \( \sum_{n=1}^\infty a_n \). Each term in the series is derived from a function of the variable \( n \), known as the sequence. Working with infinite series goes beyond just adding numbers; it involves understanding patterns and behaviors of sequences.
  • Infinite series can converge or diverge, which informs us about the behavior as more terms are added.
  • The series converges if its sequence of partial sums has a limit, otherwise, it diverges.
  • Manipulating series often requires algebraic or calculus-based techniques for simplification and convergence testing.
Example applications include power series and Fourier series, which play critical roles in mathematical analysis and applied fields. Understanding infinite series lays the groundwork for solving differential equations, evaluating limits, and computing mathematical approximations.

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