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Let αbe any real number. Show that there is a rearrangement of the terms of the alternating harmonic series that converges to α. (Hint: Argue that if you add up some finite number of the terms of k=112k1, the sum will be greater than α. Then argue that, by adding in some other finite number of the terms of

k=112k , you can get the sum to be less than α. By alternately adding terms from these two divergent series as described in the preceding two steps, explain why the sequence of partial sums you are constructing will converge to α.)

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01

Step 1. Given information.

given,

02

Step 2. 

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