Use the Alternating Series Test, if applicable, to determine the convergence or divergence of the series. ∑n=1∞(−1)nn5

Question

Use the Alternating Series Test, if applicable, to determine the convergence or divergence of the series.  ∑n=1∞(−1)nn5

Margot Mill · Accepted Answer

The given series is ∑n=1∞(−1)nn5 To check its convergence or divergence using Alternating series test. Solution: The alternating series says that if we have series of form, ∑n=1∞(−1)nbn, then if, 1)limn→∞bn=0 and 2)bn is a decreasing sequence, then the series ∑n=1∞(−1)nbn is said to be convergent. Since we have series ∑n=1∞(−1)nn5 we have sequence bn as bn=1n5, Now to check both the condition using alternating series test for convergence, 1) limn→∞bn=limn→∞1n5=0 2) n5&amp;lt;(n+1)5 1n5&amp;gt;1(n+1)5 bn&amp;gt;bn+1 So bn is the decreasing sequence as well, Since , both the condition are satisficed so the given series is convergent. Hence, the given series ∑n=1∞(−1)nn5 is convergent.

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

Use the Alternating Series Test, if applicable, to determine the convergence or divergence of the series. sum_{n=1}^inftyfrac{(-1)^n}{n^5}

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