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In part (a), to show the convergence of the series we checked that is decreasing and positive. The limit of this sequence is equal to , because by applying l'Hopital Rule:
Thus by the Alternating Series Test, this series is convergent.
For a convergent Alternating Series, as mentioned in hint (1), we would like to approximate the sum with the sum of the first five terms as the following
By the error formula for Alternating Series, the upper bound for the error of approximation is the immediate term of the sequence after which is
This can be justified by the following argument:
Let's write the error term in the expanded form:
By playing with brackets we can find the bound for the error. First:
due to the decreasing terms of the sequence, the sum of the terms in each of the ( ) will be a negative number, so
This time,
Again because of decreasing terms, the sum of the terms in each ( ) is positive
(positive terms)
so , which implies
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