Solution we have done all the work already we know

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solution. We have done all the work already. We know the series converges since it is alternating. However k =1 | ( - 1) k - 1 1 k | = k =1 1 k diverges since it is the harmonic series (i.e. a p -series with p = 1). So the convergence of k =1 ( - 1) k - 1 1 k is conditional. Conditionally convergent series have the following remarkable property: Given any real number L R , it is possible to rearrange the terms of the series so that the new series converges to L . On the other hand, this is not true for absolutely convergent series: proposition 3.3. Let k =1 a n = L converge absolutely, and let k =1 b n be a series whose terms are obtained by rearranging the terms of k =1 a n . Then k =1 b n = L . 3
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  • Fall '08
  • HIMONAS
  • Calculus, Mathematical Series, Mathematical analysis, Leven

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