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Calculus IIChapter 4: Series and SequencesSection 4-14

Power Series — Question 3

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Question 3

Find the radius and interval of convergence of ∑n=1∞xn/n\displaystyle\sum_{n=1}^{\infty}x^n/n. Identify the harmonic or alternating harmonic series arising at each endpoint.

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Question 3 – Solution

Step 1: Find the interior interval.

|xn+1/(n+1)xn/n|=|x|nn+1→|x|.\left|\frac{x^{n+1}/(n+1)}{x^n/n}\right|=|x|\frac{n}{n+1}\longrightarrow|x|. The Ratio Test gives convergence for |x|<1|x|<1 and divergence for |x|>1|x|>1, so R=1R=1.

Step 2: Test x=1x=1.

The series becomes ∑1/n\sum1/n, the divergent harmonic series.

Step 3: Test x=−1x=-1.

The series becomes ∑(−1)n/n\sum(-1)^n/n, which converges by the Alternating Series Test (but not absolutely).

Conclusion.

The interval is [−1,1)\boxed{[-1,1)}. Convergence at the left endpoint is conditional.

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