(ii) |
If  is an absolutely convergent series, and any rearrangement of the series does not affect its convergence or its sum. However, this is not the case with an alternating series. In fact, if a alternating series is convergent, then by a suitable rearrangement, it can be made to converge to any given real numbers. For more elaboration reader may consult any book on Real Analysis. |
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PRACTICE EXCERCISES |
1. |
Show that the following alternating series are convergent. |
(i) |
 .
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(ii) |
 . |
(iii) |
 . |
(iv) |
 .
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2. |
Show that the following alternating series are absolutely convergent |
(i) |
 . |
(ii) |
 .
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(iii) |
 . |
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