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Home Infinite Series Absolute and Conditional Convergence Theorem 2  
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Theorem 2
Given a series one can form a new series by listing the terms in a different order, for example a_{1} + a_{3} + a_{2} + a_{5} + a_{4} + .... Such a series is called a rearrangement of . The difference between absolute convergence and conditional convergence is shown emphatically by the following pair of theorems. THEOREM 2 A. Every rearrangement of an absolutely convergent series is also convergent and has the same sum. B. Let be a conditionally convergent series. (i) The series has a rearrangement which diverges to ∞. (ii) The series has another rearrangement which diverges to ∞. (iii) For each real number r, the series has a rearrangement which converges to r. We shall not prove these theorems. Instead we give a pair of rearrangements of the conditionally convergent series one diverging to ∞ and the other converging to  1. The alternating series conditionally converges to a number between ½ and 1. To get a rearrangement which diverges to ∞, we write down terms in the following order:
We thus obtain the series Each block of 2^{m} positive terms adds up to at least ¼, However, all the negative terms except 1/2 and ¼ have absolute value ≤ 1/6. Hence after the mth negative term the partial sum is more than Therefore the partial sums, and hence the series, diverge to x. To get a rearrangement which converges conditionally to 1 we proceed as follows: Write down negative terms until the partial sum is below 1, then positive terms until the partial sum is above 1, then negative terms until the partial sum is below 1, and so on. The mth time the partial sum goes above 1, it must be between 1 and  1 + (1/m). The with time it goes below 1 it must be between 1 and 1 (1/m). Therefore the series converges to 1.


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