Math Problem Statement
consider sequence 1+1/(13)+1/(135)++1/(1357)+...++1/(13...*(2n-1)). Is it bounded above?
Solution
To determine whether the given sequence is bounded above, let's analyze the series step by step.
Sequence Description:
The series is
The -th term of the sequence is
Step 1: Bounding Each Term
The -th term can be written as: where is the double factorial, representing the product of all odd integers from to .
For large , the growth of the double factorial is rapid. Specifically, it can be approximated as:
This shows that decreases very quickly, indicating convergence of the series.
Step 2: Sum of the Series
We need to check if the series sum, , converges or diverges.
To compare, note that:
The series converges because factorial terms grow extremely fast. Therefore, by the comparison test, the given series also converges.
Step 3: Bounded Above
Since the series converges, its partial sums, , approach a finite value as . Therefore, the sequence is bounded above by this finite value .
Conclusion:
The sequence of partial sums, , is bounded above because the series converges to a finite value.
Would you like a deeper dive into factorial approximations, convergence tests, or other related concepts?
Related Questions:
- How does the factorial grow compared to exponential functions?
- What is the difference between single and double factorials?
- Can we compute an exact value for this series?
- How do we use the ratio test to confirm convergence?
- How is the comparison test applied in series convergence?
Tip: For rapidly decreasing series, comparing to simpler series like or geometric series often helps establish convergence.
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Math Problem Analysis
Mathematical Concepts
Sequences and Series
Double Factorials
Convergence of Series
Comparison Test
Formulas
a_n = 1 / (1 * 3 * 5 * ... * (2n-1))
(2n-1)!! = Product of odd integers from 1 to (2n-1)
Convergence criteria using Comparison Test
Theorems
Comparison Test for Convergence
Factorial Approximation
Suitable Grade Level
Grades 11-12 and above (Advanced High School or College-level Mathematics)
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