*Solutions to previous blog questions on harmonic series are presented below:*

**Basic Reference: Popular Problems and Puzzles in Mathematics by Asok Kumar Mallik, IISc Press, Foundation Books; Amazon India link:**

https://www.amazon.in/Popular-Problems-Puzzles-Mathematics-Mallik/dp/938299386X/ref=sr_1_2?ie=UTF8&qid=1505622919&sr=8-2&keywords=popular+problems+and+puzzles+in+mathematics

**Detailed Reference: **

Mallik, A. K. 2007: “Curious consequences of simple sequences,” **Resonance, **(January), 23-37.

*Personal opinion only: Resonance is one of the best Indian magazines/journals for elementary/higher math and physics. It behooves you to subscribe to it. It will help in RMO, INMO and Madhava Mathematics Competition of India.*

http://www.ias.ac.in/Journals/Resonance_–_Journal_of_Science_Education/

**Solutions:**

- The thirteenth century French polymath Nicolas-Oresme proved that the harmonic series : does not converge. Prove this result.

**Solution 1:**

Nicolas Oreme had provided a simple proof as it involves mere grouping of terms, noticing patterns and making comparisons:

Therefore, diverges as we go on adding one half indefinitely. Here is another way to prove this:

Consider

By multiplying and dividing both sides by 2 and then by regrouping the terms, we get:

leading to a contradiction that The contradiction arose because only finite numbers remain unaltered when multiplied and divided by 2. So, is not a finite number, that is, it diverges.

2. Prove that does not converge.

**Solution 2:**

.

Since diverges, so does .

3. Prove that does not converge.

**Solution 3:**

diverges as each term in this series is greater than the corresponding term of , which we have just sent to diverge.

*Cheers,*

Nalin Pithwa.

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