Question

Show that there is a rearrangement of the terms of the alternating harmonic series that diverges to $\infty$. (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 $1$. 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 $1$. Next, explain why you can get the sum to be greater than $2$if you add in more terms of$\sum _{k=1}^{\mathrm{\infty }} \frac{1}{2k-1}$. Then argue again that you can get the sum to be less than $2$by adding in more terms of $\sum _{k=1}^{\mathrm{\infty }} \left(-\frac{1}{2k}\right)$. By alternately adding terms from these two divergent series in a similar fashion as just described, explain why the sum can then go above $3$, then below $3$, above $4$, then below $4$, etc. Explain why such a process results in a sequence of partial sums that diverges to $\infty$.

Short answer

Ans:

Step-by-step solution

Step 1. Given information.

given,

Step 2.