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Calculus IIChapter 4: Series and SequencesSection 4-8

Alternating Series Test — Question 10

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Question 10

Consider ∑n=1∞(−1)nnn2+1\displaystyle\sum_{n=1}^{\infty}(-1)^n\frac{n}{n^2+1}.

  1. Use a derivative to verify decreasing magnitudes.

  2. Apply the AST.

  3. Test absolute convergence and classify the series.

Original worksheet page 1: question and worked solution for 4-8-010
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Question 10 – Solution

Step 1: Verify monotonicity.

Let f(x)=x/(x2+1)f(x)=x/(x^2+1). Then f′(x)=1−x2(x2+1)2≤0(x≥1).f'(x)=\frac{1-x^2}{(x^2+1)^2}\le0\qquad(x\ge1). Thus bn=n/(n2+1)b_n=n/(n^2+1) is nonincreasing and positive.

Step 2: Check the limit and apply AST.

bn=1/n1+1/n2→0.b_n=\frac{1/n}{1+1/n^2}\longrightarrow0. Therefore the alternating series converges.

Step 3: Test absolute convergence.

limn→∞n/(n2+1)1/n=limn→∞n2n2+1=1.\lim_{n\to\infty}\frac{n/(n^2+1)}{1/n} =\lim_{n\to\infty}\frac{n^2}{n^2+1}=1. By comparison with the divergent harmonic series, the absolute series diverges. Hence convergence is conditional.

Original worksheet page 2: question and worked solution for 4-8-010

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