91Ó°ÊÓ

Show that there is a rearrangement of the terms of the alternating harmonic series that diverges to $∞$. (Hint: Argue that if you add up some finite number of the terms of $\underset{k=1}{\overset{\mathrm{∞}}{∑}} \frac{1}{2k−1}$, the sum will be greater than $1$. Then argue that, by adding in some other finite number of the terms of $\underset{k=1}{\overset{\mathrm{∞}}{∑}} \left(−\frac{1}{2k}\right)$, 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$\underset{k=1}{\overset{\mathrm{∞}}{∑}} \frac{1}{2k−1}$. Then argue again that you can get the sum to be less than $2$by adding in more terms of $\underset{k=1}{\overset{\mathrm{∞}}{∑}} \left(−\frac{1}{2k}\right)$. 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 $∞$.

Step by step solution

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