Math Problem Statement
Solution
To find the radius of convergence of the given series
we use the root test. Here's the solution step-by-step:
Step 1: Root Test Formula
The root test states that for the series
we compute
- If , the series converges absolutely.
- If , the series diverges.
- If , the test is inconclusive.
For the series , the radius of convergence is determined by:
Step 2: Identify
Here, . We focus on since the series includes .
Step 3: Apply the Root Test
We compute:
[ \sqrt[n]{|a_n|} = \sqrt[n]{\left(\frac{n-1}{2n+4}\right)^n} = \frac{n-1}{2n+4}. ]
Next, we take the limit superior () as :
Simplify the fraction:
As , and , so:
Step 4: Account for
The series includes , so we now include in the root test. For convergence, we require:
This simplifies to:
Thus, the radius of convergence is:
Final Answer:
The radius of convergence is:
Would you like more details or have any questions?
Related Questions:
- How is the root test different from the ratio test for convergence?
- Can you explain why is used instead of a standard limit in the root test?
- What would change if the series included a factorial term in ?
- How would you determine the interval of convergence from the radius of convergence?
- How does absolute convergence differ from conditional convergence?
Tip:
Always simplify the fraction inside the step carefully, as it determines the result for the radius of convergence.
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Math Problem Analysis
Mathematical Concepts
Series Convergence
Root Test
Radius of Convergence
Formulas
Root Test: L = \limsup_{n \to \infty} \sqrt[n]{|a_n|}
Radius of Convergence: \frac{1}{R} = \limsup_{n \to \infty} \sqrt[n]{|a_n|}
Theorems
Root Test
Suitable Grade Level
Undergraduate Mathematics
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