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JH

# Prove that if $\sum_n$ is a conditionally convergent series and $r$ is any real number, then there is a rearrangement $\sum a_n$ whose sum is $r.$ [Hints: Use the notation of Exercise 51. Take just enough positive terms $a_{n}^{+}$ so that their sum is greater than $r$. Then add just enough negative terms $a_{n}^{-}$ so that the cumulative sum is less than $r$. Continue in this manner and use Theorem 11.2.6.]

## Thus, for all $n>0,0<\left|a_{\sigma(n)}-M\right|<\left|a_{k}\right| .$ So the following is true:$\sum_{n=1}^{\infty} a_{\sigma(n)}=M .$ (similar for $M \leq 0 )$

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##### Kristen K.

University of Michigan - Ann Arbor

##### Michael J.

Idaho State University

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##### Kristen K.

University of Michigan - Ann Arbor

##### Michael J.

Idaho State University

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Join Bootcamp