Math Problem Statement
Solution
The problem is to determine the convergence or divergence of the given series:
If it converges, calculate the sum.
Step 1: Analyze the series for convergence
The series can be simplified using partial fraction decomposition. Write the general term:
Solve for and :
Expanding the numerator:
Equating coefficients:
From , . Substituting into the second equation:
Thus, the decomposition is:
Step 2: Rewrite the series
The series becomes:
This is a telescoping series. Expanding the first few terms:
Most terms cancel out, leaving:
Step 3: Determine convergence and calculate the sum
As , . Thus, the series converges, and the sum is:
Final Answer:
The series converges, and its sum is:
Would you like further clarification or details? Here are related questions to explore:
- What is the significance of partial fraction decomposition in series analysis?
- How do telescoping series simplify the summation process?
- Can the same method be applied to other similar series? Give examples.
- What are other tests for convergence besides telescoping?
- How would the sum change if the starting index were different?
Tip: Always check for telescoping behavior in series involving rational functions; it often simplifies calculations dramatically!
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Math Problem Analysis
Mathematical Concepts
Series convergence
Partial fraction decomposition
Telescoping series
Formulas
Partial fraction decomposition: 2/((n-1)(n+1)) = A/(n-1) + B/(n+1)
Telescoping series simplification: ∑(n=a to ∞)(Tn - T(n+1)) = Limit of remaining terms as n → ∞
Theorems
Telescoping series property: Most terms cancel out, leaving only boundary terms.
Suitable Grade Level
Undergraduate Calculus
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