Problem 5
Question
In Exercises \(1-8,\) use the Ratio Test to determine if each series conyerges ahsolutely or diveroes. $$\sum_{n=1}^{\infty} \frac{n^{4}}{(-4)^{n}}$$
Step-by-Step Solution
Verified Answer
The series converges absolutely by the Ratio Test.
1Step 1: Identify the terms of the series
The general term of the series given is \( a_n = \frac{n^4}{(-4)^n} \). Our goal is to apply the Ratio Test to this series.
2Step 2: Apply the Ratio Test formula
The Ratio Test states that for the series \( \sum a_n \), compute \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \). If \( L < 1 \), the series converges absolutely; if \( L > 1 \), it diverges; if \( L = 1 \), the test is inconclusive.
3Step 3: Calculate \( a_{n+1} \)
For \( a_n = \frac{n^4}{(-4)^n} \), the next term \( a_{n+1} = \frac{(n+1)^4}{(-4)^{n+1}} \).
4Step 4: Form the ratio \( \frac{a_{n+1}}{a_n} \)
Substitute \( a_{n+1} \) and \( a_n \) into the ratio: \[ \frac{a_{n+1}}{a_n} = \frac{\frac{(n+1)^4}{(-4)^{n+1}}}{\frac{n^4}{(-4)^n}} = \frac{(n+1)^4}{n^4} \times \frac{(-4)^n}{(-4)^{n+1}} = \frac{(n+1)^4}{4n^4} \].
5Step 5: Evaluate the limit of the ratio
Calculate \( L = \lim_{n \to \infty} \left| \frac{(n+1)^4}{4n^4} \right| \). Simplifying: \[ \frac{(n+1)^4}{4n^4} = \frac{(n^4 + 4n^3 + 6n^2 + 4n + 1)}{4n^4} \approx \frac{n^4}{4n^4} = \frac{1}{4} \]. Thus, \( L = \frac{1}{4} \).
6Step 6: Interpret the result of the Ratio Test
Since \( L = \frac{1}{4} < 1 \), the Ratio Test indicates that the series \( \sum_{n=1}^{\infty} \frac{n^4}{(-4)^n} \) converges absolutely.
Key Concepts
series convergenceabsolute convergencelimit calculation
series convergence
A series is a sum of terms of a sequence. When we talk about series convergence, we're trying to determine if the sum approaches a specific value as we add more and more terms. For a series to converge, the terms must get smaller and closer to zero as we go further along the sequence. If they do not, the sum can grow indefinitely, meaning the series diverges.
Mathematically speaking, a series converges when its sequence of partial sums approaches a finite limit. This is crucial because it allows us to assign a concrete value to the sum of infinitely many terms, which is a foundation for calculus and analysis.
Several tests help determine if a series converges, such as the Ratio Test, which we focus on here. A series can even converge "absolutely", meaning the sum of the absolute values of the terms converge. Absolute convergence is a stronger condition than simple convergence.
Mathematically speaking, a series converges when its sequence of partial sums approaches a finite limit. This is crucial because it allows us to assign a concrete value to the sum of infinitely many terms, which is a foundation for calculus and analysis.
Several tests help determine if a series converges, such as the Ratio Test, which we focus on here. A series can even converge "absolutely", meaning the sum of the absolute values of the terms converge. Absolute convergence is a stronger condition than simple convergence.
absolute convergence
Absolute convergence is an essential concept when discussing series. A series is absolutely convergent if the series of absolute values converges. For example, the series \[\sum_{n=1}^{\infty} \left| a_n \right|\]converges.
Why is this important? Because if a series converges absolutely, it indicates a stronger form of convergence. This means the series will converge even if we change the order of its terms, which isn't always true for simply convergent series.
The Ratio Test is often used to test for absolute convergence. If applying the Ratio Test gives a limit less than 1, the series is absolutely convergent. This guarantee allows mathematicians to manipulate and work with series more flexibly, knowing their values won't unexpectedly change due to rearrangements.
Why is this important? Because if a series converges absolutely, it indicates a stronger form of convergence. This means the series will converge even if we change the order of its terms, which isn't always true for simply convergent series.
The Ratio Test is often used to test for absolute convergence. If applying the Ratio Test gives a limit less than 1, the series is absolutely convergent. This guarantee allows mathematicians to manipulate and work with series more flexibly, knowing their values won't unexpectedly change due to rearrangements.
limit calculation
Calculating limits is a fundamental task when analyzing series, and it's an essential part of the Ratio Test. The Ratio Test involves calculating the limit:
\[L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\]
A limit is a value that a sequence or function approaches as the input approaches some point. In the context of the Ratio Test, this limit helps us assess the behavior of the terms of the series as they go to infinity.
For a series, if this limit, \(L\), is less than 1, it suggests that the terms are shrinking rapidly enough to ensure the overall sum stays bounded—thus, the series converges absolutely. Conversely, if \(L\) is greater than 1, the terms are not getting small quickly enough, leading the series to diverge. Evaluating such limits requires good algebraic manipulation skills, and often these limits can be simplified using dominant terms in polynomials or rational expressions.
\[L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\]
A limit is a value that a sequence or function approaches as the input approaches some point. In the context of the Ratio Test, this limit helps us assess the behavior of the terms of the series as they go to infinity.
For a series, if this limit, \(L\), is less than 1, it suggests that the terms are shrinking rapidly enough to ensure the overall sum stays bounded—thus, the series converges absolutely. Conversely, if \(L\) is greater than 1, the terms are not getting small quickly enough, leading the series to diverge. Evaluating such limits requires good algebraic manipulation skills, and often these limits can be simplified using dominant terms in polynomials or rational expressions.
Other exercises in this chapter
Problem 5
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