Interval of Convergence
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The interval of convergence for the power series ∑(x^n/n*3^n) is [-3, 3) based on applying the Ratio Test. The Radius of Convergence is 3, as determined by the Ratio Test limiting the values of x between -3 and 3.
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Interval of Convergence
- 1. Interval and Radius of Convergence A Practice Problem
- 2. State the interval and the radius of convergence of the power series below: • 𝑛=1 ∞ 𝑥 𝑛 𝑛∙3 𝑛
- 3. Step 1: • Use the Ratio Test! • lim 𝑛→∞ 𝑥 𝑛+1 (𝑛+1)3 𝑛+1 ∙ 𝑛3 𝑛 𝑥 𝑛 • = lim 𝑛→∞ 𝑥𝑛 3(𝑛+1) • = 𝑥 3 < 1 • -3 < x < 3
- 4. Step 2: • Test the endpoints! • At x = -3: 𝑛=1 ∞ (−1) 𝑛 𝑛 ; this series converges by the AST • At x = 3: 𝑛=1 ∞ 1 𝑛 ; this series diverges because it is the harmonic series
- 5. Step 3: • Write the interval! • [-3, 3)
- 6. Step 4: • Identify the length of the radius of convergence! • R = 3