Math Problem Statement
Find the interval, I, of convergence of the series. (Enter your answer using interval notation.)
Solution
The given series is:
To find the interval of convergence, we can use the Ratio Test. The Ratio Test states that for a series , the series converges if the limit
satisfies .
Step 1: Define the general term
The general term of the series is:
Step 2: Compute the ratio
Simplifying:
Step 3: Compute the limit
For large , behaves like . Therefore:
Since :
Step 4: Apply the Ratio Test
For the series to converge, we need:
This implies:
Multiplying through by 5:
Subtracting 5 from all parts:
Step 5: Test the endpoints
-
At : The series is an alternating series where the terms approach 0 as increases, but does not decrease fast enough to ensure convergence by the Alternating Series Test or the Integral Test. Therefore, it diverges.
-
At : The series also diverges because it resembles a harmonic series, which diverges.
Conclusion
The series converges for .
Thus, the interval of convergence is:
Would you like further details or have any questions?
Here are some related questions you might consider:
- What is the Ratio Test and how does it determine convergence?
- How can the Root Test be applied to determine convergence?
- What are the endpoints of convergence and how are they tested?
- How does the Alternating Series Test work for testing convergence?
- What is the behavior of logarithmic functions at large values of ?
- What happens if the series has absolute convergence within the interval?
- How does the Integral Test help in determining the convergence of a series?
- What are some practical examples of series that involve logarithmic terms?
Tip: When checking the endpoints of convergence, always consider applying tests like the Alternating Series Test or Integral Test to determine if they are included in the interval.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Series convergence
Ratio Test
Interval notation
Formulas
-
Theorems
Ratio Test
Suitable Grade Level
Advanced Undergraduate
Related Recommendation
Find Interval of Convergence for Power Series with Given Ratio Limit
Analyzing Series Convergence: Ratio Test and Limit Analysis
Determine Radius and Interval of Convergence of Power Series Using Ratio Test
Find Interval of Convergence Using Different Tests
Interval of Convergence for Series: ∑n=1∞(3x−1)n/(n3+n)