Math Problem Statement
Solution
We are asked to use the Limit Comparison Test to determine whether the series
converges or diverges.
Step 1: Simplify the given term
First, simplify the general term of the series:
As becomes large, behaves like since the becomes negligible. So, for large :
Thus, for large , the general term behaves like:
Step 2: Choose a comparison series
The simplified term behaves like , which suggests we compare the given series with the harmonic series:
We know that the harmonic series diverges.
Step 3: Apply the Limit Comparison Test
Let and . The Limit Comparison Test says that if
where , then both series either converge or diverge together.
Now, compute the limit:
Since the limit is a finite positive number (1), the Limit Comparison Test tells us that the given series behaves like the harmonic series, which diverges.
Conclusion:
The series diverges.
Would you like further details or have any questions about the solution?
Here are five related questions to explore further:
- What are the conditions for the Limit Comparison Test to be applicable?
- How do you handle cases where the Limit Comparison Test gives or ?
- How can we determine the behavior of a series if the comparison is inconclusive?
- What are other common methods to test for series convergence besides the Limit Comparison Test?
- What are the differences between the Limit Comparison Test and the Direct Comparison Test?
Tip: Always simplify the general term in a series as much as possible before deciding on the appropriate test for convergence.
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Math Problem Analysis
Mathematical Concepts
Limit Comparison Test
Convergence
Series
Formulas
√((k^2 - k) / k^4) ≈ 1 / k for large k
Limit Comparison Test: lim(k→∞) (a_k / b_k) = L, where 0 < L < ∞
Theorems
Limit Comparison Test
Harmonic Series Divergence
Suitable Grade Level
College Level (Calculus II or higher)
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