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This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part. Tutorial Exercise Test the series for convergence or divergence. 6 7 − 6 9 + 6 11 − 6 13 + 6 15 −

User Mpkuth
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Step 1: Identify the series

The series in question is an alternating series in the form ∑(-1)^n * a_n for n=0 to infinity where each a_n = 6/(2n + 7).

Step 2: Check whether the series satisfies conditions for the Alternating Series Test

The two conditions to be met for the Alternating Series Test are
(a) the magnitudes of the terms of the series, |a_n|, must be decreasing and
(b) the limit as n approaches infinity for a_n must be zero

(Condition a): To check if the terms |a_n| = 6/(2n + 7) are decreasing, we take two consecutive terms a_n1 = 6/(2n + 7) and a_n2 = 6/(2(n+1) + 7) = 6/(2n + 9). For |a_n1| > |a_n2|, it results into (2n + 9) > (2n + 7), which is true. Hence, the terms, |a_n|, are indeed decreasing.

(Condition b): Now, we check if the limit of a_n = 6/(2n + 7) as n approaches infinity is zero. The limit is indeed zero as the denominator grows faster than the numerator as n increases.

Step 3: Determine whether the series converges or diverges

Since the series meets both conditions of the Alternating Series Test -- the terms are decreasing in magnitude and the limit of the terms a_n as n approaches infinity is zero -- we can conclude that the series converges.

In conclusion, the alternating series 6/7 - 6/9 + 6/11 - 6/13 + 6/15... converges.

User DarkDeny
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