$$
\sum_{n=1}^{\infty}(-1)^{n+1} \frac{n}{n^{2}+1}
$$

Does the series converge conditionally, converge absolutely, or diverge?

Question

$$
\sum_{n=1}^{\infty}(-1)^{n+1} \frac{n}{n^{2}+1}
$$

Does the series converge conditionally, converge absolutely, or diverge?

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To determine the convergence of the series, we first consider the alternating series test. ‖ ‖ step 2 ⋮ The alternating series test states that if the absolute value of the terms $ a_n = \frac{n}{n^2+1} $ is decreasing and $ \lim_{n \to \infty} a_n = 0 $, then the series $ \sum_{n=1}^{\infty} (-1)^{n+1} a_n $ converges. ‖ ‖ step 3 ⋮ We show that $ a_n $ is decreasing by showing that $ a_{n+1} < a_n $ for all $ n $. ‖ ‖ step 4 ⋮ We calculate the limit $ \lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{n}{n^2+1} $ and find that it equals 0. ‖ ‖ step 5 ⋮ Since both conditions of the alternating series test are met, the series converges. ‖ ‖ step 6 ⋮ To determine if the series converges absolutely, we consider the series $ \sum_{n=1}^{\infty} |(-1)^{n+1} a_n| = \sum_{n=1}^{\infty} \frac{n}{n^2+1} $. ‖ ‖ step 7 ⋮ We use the comparison test, comparing $ \frac{n}{n^2+1} $ with $ \frac{1}{n} $, and note that $ \frac{n}{n^2+1} $ is smaller for all $ n \geq 1 $. ‖ ‖ step 8 ⋮ Since $ \sum_{n=1}^{\infty} \frac{1}{n} $ diverges (harmonic series), and $ \frac{n}{n^2+1} $ is smaller, we cannot conclude that $ \sum_{n=1}^{\infty} \frac{n}{n^2+1} $ converges. ‖ ‖ step 9 ⋮ Therefore, we cannot say that the series converges absolutely. ‖ ‖ step 10 ⋮ The series $ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{n}{n^{2}+1} $ converges conditionally based on the alternating series test. ‖ ∻Answer∻ ⚹ The series converges conditionally. ⚹ ∻Key Concept∻ ⚹Alternating Series Test and Absolute Convergence⚹ ∻Explanation∻ ⚹The series passes the alternating series test and thus converges, but it does not converge absolutely because the absolute series does not pass the comparison test with the harmonic series.⚹◊What is the alternating series test and how can it be applied to the given series?⍭ Generate me a similar question◊

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