Absolute convergent, conditionally convergent or divergent, Mathematics

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Find out if each of the subsequent series are absolute convergent, conditionally convergent or divergent.

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Solution:

(a)

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The above is the alternating harmonic series and we saw in the previous section that it is a convergent series so we there is no requirement to check that here.  Thus, let's see if it is an absolutely convergent series.  To do this we will need to check the convergence of.

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This is harmonic series and we were familiar from the integral test part that it is divergent. Hence, this series is not absolutely convergent.  Though, It is conditionally convergent as the series itself does converge.

(b) 

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In this example let's just test absolute convergence first as if it's absolutely convergent we won't need to difficulty checking convergence as we will get that for free.

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This type of series is convergent by the p-series test and thus the series is absolute convergent. 

Note: This does say as well that it is a convergent series.


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