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In this exercise, we examine one of the conditions of the Alternating Series Test. Consider the...

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Basic Math

In this exercise, we examine one of the conditions of theAlternating Series Test. Consider the alternating series

1−1+1/2−1/4+1/3−1/9+1/4−1/16+⋯,

where the terms are selected alternately from the sequences{1/n} and {−1/n^2}.

  1. Explain why the nth term of the given series converges to 0 as ngoes to infinity.

  2. Rewrite the given series by grouping terms in the followingmanner:

    (1−1)+(1/2−1/4)+(1/3−1/9)+(1/4−1/16)+⋯.

    Use this regrouping to determine if the series converges ordiverges.

  3. Explain why the condition that the sequence {an}{an}decreases to a limit of 0 is included in the AlternatingSeries Test.

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