Question

What is the first test you should use in analyzing the convergence of a series?

Short answer

Question: Using the Test for Divergence, determine if the series converges or diverges: \(\sum_{n=1}^\infty \dfrac{n^2}{n^2 + 1} \)

Step-by-step solution

Write the general term of the series

Firstly, it's necessary to have the general term of the series to be analyzed, which is usually represented as a_n or a(n).

Determine the limit of the general term as n goes to infinity

Calculate the limit of the general term (a_n) as n approaches infinity. To do this, use the limit properties and techniques such as factoring, L'Hopital's Rule (if applicable), rationalizing, etc.

Examine the result of the limit

Analyze the outcome of the limit calculated in Step 2. If the limit is not equal to zero (\(\lim_{n \to \infty} a_n \neq 0\)), then the series is divergent. If the limit is zero (\(\lim_{n \to \infty} a_n = 0\)), the Test for Divergence is inconclusive, and further tests (such as the Integral Test, Comparison Test, or Ratio Test) are needed to determine if the series converges or diverges.

Key concepts

Test for Divergence

The Test for Divergence is often the starting point when analyzing the convergence of a series. It's a simple and straightforward method, ideal for initial assessments. In this test, we focus on the general term of the series, typically denoted as \(a_n\). The main goal here is to determine the behavior of the term \(a_n\) as the index \(n\) approaches infinity.

• Calculate the limit: First, compute the limit \(\lim_{n \to \infty} a_n\).
• Analyze the result: This test centers around checking the result of the limit.

If this limit does not equal zero, the series is divergent. This means:

• If \(\lim_{n \to \infty} a_n eq 0\), the terms are not approaching zero, so the series is clearly divergent.
• If \(\lim_{n \to \infty} a_n = 0\), the test for divergence is inconclusive, meaning we cannot say definitively whether the series converges or diverges.

In case the test for divergence is inconclusive, we need to employ other methods to further investigate the series' behavior.

Limit of a Sequence

The concept of limits is crucial in understanding both sequences and series. A sequence is an ordered list of numbers, defined by a general term \(a_n\). As \(n\), the index, increases indefinitely, we reach the concept of the limit of a sequence.

• The importance of limits: The limit helps us understand the long-term behavior of \(a_n\) as \(n\) reaches infinity.
• Calculation: To find \(\lim_{n \to \infty} a_n\), we apply several limit-solving strategies like algebraic simplification, factoring, or L'Hopital's Rule to evaluate the outcome.

A well-defined limit provides valuable insights:

• If \(\lim_{n \to \infty} a_n = L\) where \(L\) is finite, the sequence approaches \(L\) as \(n\) becomes large.
• If the limit does not exist or is infinite, the sequence doesn't stabilize, indicating divergence in the specific sense of sequences.

Understanding limits is foundational in the analysis of series, setting the stage for more complex convergence tests.

Convergence Tests

When the Test for Divergence fails to provide a conclusive answer, several convergence tests help determine the behavior of a series. These tests are essential tools for further assessment:

• Integral Test: This test relates a series to an integral to determine convergence or divergence. If the integral of \(f(x)\), the continuous counterpart of \(a_n\), is convergent, the series is convergent.
• Comparison Test: This involves comparing the series with another series whose convergence is known. If \(a_n \leq b_n\) and the series \(b_n\) converges, then \(a_n\) converges. Conversely, if \(a_n \geq c_n\) and \(c_n\) diverges, then \(a_n\) diverges.
• Ratio Test: Here, we look at the limit of the ratio of successive terms. If \(\lim_{n \to \infty} \frac{a_{n+1}}{a_n} < 1\), the series is absolutely convergent.

Choosing the right test depends on the series' form and characteristics. Each test provides a specific lens through which to scrutinize the infinite sum of terms, ensuring a comprehensive understanding of convergence.