Rearrangement
True or False ?
1 − 2 1 + 3 1 − 4 1 + 5 1 − 6 1 + 7 1 − 8 1 + 9 1 − 1 0 1 + 1 1 1 − 1 2 1 + ⋯ = ( 1 − 2 1 − 4 1 ) + ( 3 1 − 6 1 − 8 1 ) + ( 5 1 − 1 0 1 − 1 2 1 ) + ⋯
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2 solutions
Why is it that the sum of an infinite series is the same no matter how you rearrange its terms, only if the series is absolutely convergent?
This is an example of Riemann's Rearrangement Theorem
The terms of an infinite series can only be rearranged, and still be considered equal, when the series is absolutely convergent. The alternating harmonic series is not absolutely convergent because the sum of the absolute value of its terms, the harmonic series, is well known to diverge. Thus if you change the order of the terms in the alternating harmonic series, the series can converge to a different value. (In fact it's possible to make it converge to any value you'd like.)