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

In this exercise, we examine one of the conditions of the Alternating 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}.

Explain why the nth term of the given series converges to 0 as n goes to infinity.
Rewrite the given series by grouping terms in the following manner:

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

Use this regrouping to determine if the series converges or diverges.
Explain why the condition that the sequence {an}{an} decreases to a limit of 0 is included in the Alternating Series Test.

Solutions

Expert Solution

milcah answered 1 year ago

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