Math Problem Statement
Solution
To analyze the power series and determine its interval of convergence, we apply the ratio test.
Step 1: Ratio Test
The given power series is: Let .
The ratio test involves computing:
-
Compute :
-
Compute :
Simplify:
Simplify each term:
Thus:
- Take the limit as :
Step 2: Convergence
For convergence, the ratio test states :
Solve for :
This gives the interval:
Step 3: Endpoint Check
We need to check the endpoints and by substituting into the original series. This step depends on whether the series converges at these values.
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Math Problem Analysis
Mathematical Concepts
Power Series
Interval of Convergence
Ratio Test
Formulas
Ratio test: \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \)
Interval of Convergence: \( |x - c| < R \)
Theorems
Ratio Test
Suitable Grade Level
Undergraduate (Calculus or Advanced Math)
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