Abel Convergence Test

Examples 4.2.18(a):

The sum converges conditionally.

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We already know that the series does not converge absolutely (why ?). As for convergence, let us verify the conditions for Abel's test:

First, let

{ a n } = { (-1) n }
{ b n } = { 1 / n }
Then the sequence of partial sums of a n's is clearly bounded (by what number ?), and the sequence { b n } is decreasing and convergent to zero. Hence, Abel's test applies, showing that the series converges.

Therefore, the series converges conditionally.

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