Problem 30
Question
Determine whether the series is convergent or divergent. $$\sum_{n=1}^{\infty} \frac{n+5}{\sqrt[3]{n^{7}+n^{2}}}$$
Step-by-Step Solution
Verified Answer
The series is convergent.
1Step 1: Analyze the General Term
The series in question is \( \sum_{n=1}^{\infty} \frac{n+5}{\sqrt[3]{n^{7}+n^{2}}} \). The general term of the series is \( a_n = \frac{n+5}{\sqrt[3]{n^{7}+n^{2}}} \). We need to investigate whether this series converges or diverges.
2Step 2: Simplify the Expression for Large n
For large \( n \), the term \( n^7 \) dominates over \( n^2 \) in the expression \( n^7 + n^2 \). Thus, \( \sqrt[3]{n^7 + n^2} \approx \sqrt[3]{n^7} = n^{7/3} \). So the general term \( a_n \approx \frac{n}{n^{7/3}} = n^{1 - 7/3} = n^{-4/3} \).
3Step 3: Apply the p-Series Test
The simplified expression is \( a_n \approx n^{-4/3} \), which is a p-series with \( p = 4/3 \). The p-series \( \sum n^{-p} \) converges if \( p > 1 \). Here, \( 4/3 > 1 \), so the series \( \sum_{n=1}^{\infty} n^{-4/3} \) converges. Hence, the given series converges by comparison to this p-series.
Key Concepts
General Termp-Series TestSimplifying Expressions
General Term
Understanding the general term in a series is crucial for identifying its behavior. In this particular exercise, we have the series \( \sum_{n=1}^{\infty} \frac{n+5}{\sqrt[3]{n^{7}+n^{2}}} \). The general term \( a_n \) is given by \( \frac{n+5}{\sqrt[3]{n^{7}+n^{2}}} \). This term helps us decide whether the series is likely to converge or diverge by analyzing its structure. The general term acts as the building block for examining the series and provides a clearer picture of how the sum behaves as \( n \) increases.
In essence, by focusing on \( a_n \), we can simplify complex expressions and determine convergence by comparing the series with known test criteria such as the p-series test.
In essence, by focusing on \( a_n \), we can simplify complex expressions and determine convergence by comparing the series with known test criteria such as the p-series test.
p-Series Test
The p-series test is a standard method used to evaluate the convergence of series. A p-series is in the form \( \sum n^{-p} \), and its behavior depends on the value of \( p \). Specifically, a p-series converges if \( p > 1 \) and diverges if \( p \leq 1 \).
In the exercise, after simplifying the expression to match a p-series form, we find that \( a_n \approx n^{-4/3} \). Here, \( p = 4/3 \) which is greater than 1, indicating the series converges.
Understanding this test allows students to quickly ascertain the behavior of similar series. The p-series test is particularly powerful because of its simplicity and effectiveness for series that closely resemble \( n^{-p} \) structures.
In the exercise, after simplifying the expression to match a p-series form, we find that \( a_n \approx n^{-4/3} \). Here, \( p = 4/3 \) which is greater than 1, indicating the series converges.
Understanding this test allows students to quickly ascertain the behavior of similar series. The p-series test is particularly powerful because of its simplicity and effectiveness for series that closely resemble \( n^{-p} \) structures.
Simplifying Expressions
Simplifying expressions is an essential step when working with series convergence. By simplifying, you reduce complex expressions to a more manageable form, often making it easier to apply known series tests.
In this example, the original expression \( \frac{n+5}{\sqrt[3]{n^{7}+n^{2}}} \) is simplified based on the dominance of terms for large \( n \). The term \( n^7 \) is significantly larger compared to \( n^2 \), allowing the approximation of \( \sqrt[3]{n^7+n^2} \approx \sqrt[3]{n^7} = n^{7/3} \). This step simplifies \( a_n \) to \( n^{-4/3} \), a key move in identifying it as a p-series.
Simplification not only aids in applying tests like the p-series test but also helps maintain focus on the main components influencing series convergence. It transforms complex expressions into a form that is easy to handle and understand.
In this example, the original expression \( \frac{n+5}{\sqrt[3]{n^{7}+n^{2}}} \) is simplified based on the dominance of terms for large \( n \). The term \( n^7 \) is significantly larger compared to \( n^2 \), allowing the approximation of \( \sqrt[3]{n^7+n^2} \approx \sqrt[3]{n^7} = n^{7/3} \). This step simplifies \( a_n \) to \( n^{-4/3} \), a key move in identifying it as a p-series.
Simplification not only aids in applying tests like the p-series test but also helps maintain focus on the main components influencing series convergence. It transforms complex expressions into a form that is easy to handle and understand.
Other exercises in this chapter
Problem 30
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