Show that for any real number c, it is possible to rearrange the terms of the series $\sum_{n=1}^\infty (-1)^{n+1} (1/n)$ so that the sum is exactly c.
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1This is true more generally, it's in fact true for any series that converges conditionally. – Gregory Grant Nov 24 '15 at 13:13
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See here: http://mathworld.wolfram.com/RiemannSeriesTheorem.html – Gregory Grant Nov 24 '15 at 13:14
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1Reimann Rearrangement theorem – K_user Nov 24 '15 at 13:14
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The obvious places that you did not even check accumulate... https://en.wikipedia.org/wiki/Riemann_series_theorem#Examples – Did Nov 24 '15 at 13:40
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Duplicate: http://math.stackexchange.com/q/46195 – Did Nov 24 '15 at 13:42
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1Possible duplicate of Series rearrangement and Riemann's theorem – Nov 24 '15 at 23:54
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Hints:
Suppose $c>0$. Add the positive terms of the series, one by one, i.e. do the addition $1+1/3+1/5+...$ and stop till this rises above $c$.
Then, start adding the negative terms i.e. $-1/2-1/4-1/6-...$ to the above so that the result just drops below $c$.
Repeat this step with the remaining positive and negative terms.
Landon Carter
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