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:
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.
- The thirteenth century French polymath Nicolas-Oresme proved that the harmonic series : does not converge. Prove this result.
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:
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.
Since diverges, so does .
3. Prove that does not converge.
diverges as each term in this series is greater than the corresponding term of , which we have just sent to diverge.