Problem 43 - Thomas Calculus

Question

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{(-1)^{n+1}}{2 n-1} $$

Step-by-Step Solution

Answer

The sequence converges to 0.

Step 1: Clarify Sequence Behavior

The given sequence is \( a_n = \frac{(-1)^{n+1}}{2n - 1}\). We see that \((-1)^{n+1}\) alternates in sign depending on whether \(n\) is even or odd, meaning the sequence is alternating.

Step 2: Analyze Denominator Growth

As \(n\) increases, the denominator \(2n - 1\) increases linearly. This suggests that as \(n \rightarrow \infty\), the terms of the sequence get smaller, because the denominator grows without bound and influences the magnitude of the fraction.

Step 3: Consider the Alternating Nature

The numerator \((-1)^{n+1}\) causes the sequence to alternate. For even \(n\), \(a_n = \frac{1}{2n-1}\), and for odd \(n\), \(a_n = \frac{-1}{2n-1}\). Both of these approaches zero as \(n\) becomes very large.

Step 4: Establish Convergence

Even with the alternating sign, the sequence approaches 0 as the magnitude of each term decreases. Hence, for both even and odd instances, the terms approach 0 sufficiently fast, implying convergence.

Step 5: Conclusion on Limit

Since both positive and negative terms converge to 0, and given that the terms themselves oscillate in sign, the sequence actually converges to \(0\).

Key Concepts

Alternating Sequences

An alternating sequence is one where the terms change sign regularly. For example, a sequence may have terms that are positive, then negative, and continue this pattern as it progresses. This is particularly visible in the given sequence \( a_{n} = \frac{(-1)^{n+1}}{2n - 1} \) where the exponent of \((-1)^{n+1} \) dictates the sign of each term.

• For even values of \(n\), \((-1)^{n+1}\) is negative, resulting in negative terms.
• For odd \(n\), \((-1)^{n+1}\) becomes positive, resulting in positive terms.

This regular alternation in sign doesn't affect the analysis of convergence, but it does show how the sequence can hover around a central value, moving away and then back again as it progresses.

Limit of a Sequence

The limit of a sequence refers to the value that the sequence's terms approach as the variable \(n\) becomes very large. In our specific sequence, \( a_{n} = \frac{(-1)^{n+1}}{2n - 1} \), we are interested in whether there is a particular number that the terms get closer to as \(n\) increases.

• Since the denominator \(2n - 1\) grows indefinitely, the size of each term becomes smaller and smaller.
• Even though alternating sequences have terms oscillating between positive and negative values, what really matters is the size of those terms.

For the given sequence, as the calculate shows that each term tends toward \(0\) as \(n\rightarrow\infty\). This means that the limit of the sequence is indeed \(0\).

Sequence Convergence

Sequence convergence is the idea that the terms of a sequence tend to get closer to a specific value, called the limit, as \(n\) increases. If this happens, the sequence is said to converge. In the context of our sequence, \( a_{n} = \frac{(-1)^{n+1}}{2n - 1} \), the convergence analysis hinges on:

• How the denominator \(2n - 1\) increases, causing the terms to shrink.
• Despite the sign changes due to \((-1)^{n+1}\), the absolute magnitude decreases as \(n\) becomes large.

Since the terms consistently drift towards \(0\) from both above and below (due to alternating signs), we can confidently say the sequence converges to \(0\).

Behavior of Sequences

Understanding sequence behavior involves looking at how the terms change over time as \(n\) increases. For the sequence \( a_{n} = \frac{(-1)^{n+1}}{2n - 1} \), the behavior is analyzed in terms of both the direction (sign) and size (magnitude) of the terms.

• The alternating pattern shows that terms switch from positive to negative, affecting direction.
• The magnitude of each term is impacted by the linear growth of \(2n - 1\), which makes terms smaller over larger \(n\).

As seen in this sequence, while the direction oscillates, the shrinking magnitude dominates, pulling the sequence closer to \(0\). Ultimately, understanding these patterns helps us effectively determine convergence and limits.