Problem 39
Question
Determine the convergence or divergence of the sequence with the given \(n\) th term. If the sequence converges, find its limit. \(a_{n}=\frac{1+(-1)^{n}}{n}\)
Step-by-Step Solution
Verified Answer
The sequence \(a_{n} = \frac{1+(-1)^{n}}{n}\) doesn't converge, as it alternates between different limits for odd and even values of \(n\).
1Step 1: Analyze the sequence
First, it should be noticed that the sequence \(a_{n} = \frac{1+(-1)^{n}}{n}\) has two different behaviors: when \(n\) is an even number, and when \(n\) is odd.
2Step 2: Evaluate the sequence for odd and even values of n
For even values of \(n\), \(a_{n} = \frac{1+(-1)^{n}}{n} = \frac{2}{n}\), and for odd values of \(n\), \(a_{n} = \frac{1+(-1)^{n}}{n} = 0\). Thus, we actually have two subsequences here: one for which \(n\) is even, and the other for which \(n\) is odd.
3Step 3: Apply the limit properties
Now, we want to find the limit of the sequence as \(n\) goes to infinity. From step 2, it is clear that the sequence alternates between a sequence of halves: \(0, \frac{2}{2}, 0, \frac{2}{4}, 0, \frac{2}{6}, ...\). This leads to \(a_{n}\) converging to different limits based on whether \(n\) is even or odd. Thus, the sequence doesn't converge.
Key Concepts
Limit of a sequenceAlternating sequencesSubsequences
Limit of a sequence
When we talk about the limit of a sequence, we're focusing on the value that the terms in the sequence get closer to as they extend indefinitely. This often means observing what happens to the sequence as \( n \) (the term position) approaches infinity.
A sequence \( \{ a_n \} \) is said to converge to a limit \( L \) if, for every small positive number \( \epsilon \), there exists a position \( N \) such that for all terms \( n > N \), the absolute difference \( |a_n - L| < \epsilon \). Essentially, beyond a certain point, every term in the sequence should be arbitrarily close to \( L \).
If no such \( L \) exists, the sequence is considered divergent. For the sequence \( a_n = \frac{1+(-1)^n}{n} \), due to the alternating behavior, the sequence doesn't settle to just one limit. As we analyze subsequences, we find that one part converges to zero while the other part's limit changes as \( n \) progresses further.
A sequence \( \{ a_n \} \) is said to converge to a limit \( L \) if, for every small positive number \( \epsilon \), there exists a position \( N \) such that for all terms \( n > N \), the absolute difference \( |a_n - L| < \epsilon \). Essentially, beyond a certain point, every term in the sequence should be arbitrarily close to \( L \).
If no such \( L \) exists, the sequence is considered divergent. For the sequence \( a_n = \frac{1+(-1)^n}{n} \), due to the alternating behavior, the sequence doesn't settle to just one limit. As we analyze subsequences, we find that one part converges to zero while the other part's limit changes as \( n \) progresses further.
Alternating sequences
Alternating sequences are interesting because their terms swap between two behaviors, typically determined by the sign or value. In the exercise, the sequence \( a_n = \frac{1+(-1)^n}{n} \) is considered an alternating sequence since it behaves differently for odd and even \( n \).
- **Even Values**: When \( n \) is even, \( (-1)^n = 1 \), so \( a_n = \frac{2}{n} \). This tends towards zero as \( n \) increases.
- **Odd Values**: When \( n \) is odd, \( (-1)^n = -1 \), so \( a_n = 0 \).
The sequence essentially "alternates" between zero and a fraction that becomes smaller with larger \( n \). Hence, the oscillating nature of alternating sequences can sometimes prevent them from having a single limit.
- **Even Values**: When \( n \) is even, \( (-1)^n = 1 \), so \( a_n = \frac{2}{n} \). This tends towards zero as \( n \) increases.
- **Odd Values**: When \( n \) is odd, \( (-1)^n = -1 \), so \( a_n = 0 \).
The sequence essentially "alternates" between zero and a fraction that becomes smaller with larger \( n \). Hence, the oscillating nature of alternating sequences can sometimes prevent them from having a single limit.
Subsequences
A subsequence is created by selecting certain elements of a sequence, often in a regular way. Subsequence analysis helps break down complex behaviors into simpler parts.
In \( a_n = \frac{1+(-1)^n}{n} \), there are two clear subsequences:
In \( a_n = \frac{1+(-1)^n}{n} \), there are two clear subsequences:
- When \( n \) is even: the subsequence becomes \( a_{2k} = \frac{2}{2k} = \frac{1}{k} \) and converges to zero as \( k \) tends to infinity. It simplifies the process and gives a clear view of the sequence behavior when \( n \) is even.
- When \( n \) is odd: the subsequence becomes \( a_{2k+1} = 0 \), which already remains at zero, indicating convergence.
Other exercises in this chapter
Problem 38
Explain how to use the geometric series $$g(x)=\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}, \quad|x|
View solution Problem 39
Find the sum of the convergent series. $$ 1+0.1+0.01+0.001+\cdots $$
View solution Problem 39
Determine whether the series converges conditionally or absolutely, or diverges. $$ \sum_{n=0}^{\infty} \frac{\cos n \pi}{n+1} $$
View solution Problem 39
Explain how to use the geometric series $$g(x)=\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}, \quad|x|
View solution