Question

Let $\alpha$be any real number. Show that there is a rearrangement of the terms of the alternating harmonic series that converges to $\alpha$. (Hint: Argue that if you add up some finite number of the terms of $\sum _{k=1}^{\mathrm{\infty }} \frac{1}{2k-1}$, the sum will be greater than $\alpha$. Then argue that, by adding in some other finite number of the terms of

$\sum _{k=1}^{\mathrm{\infty }} \left(-\frac{1}{2k}\right)$ , you can get the sum to be less than $\alpha$. 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 $\alpha$.)

Short answer

Ans:

Step-by-step solution

Step 1. Given information.

given,

Step 2.