Famous harmonic series questions

Question 1:

The thirteenth century French polymath Nicolas Oresme proved that the harmonic series 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \ldots does not converge. Prove this result.

Question 2:

Prove that \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \ldots does not converge.

Question 3:

Prove that 1 + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \ldots does not converge.

Even if you solve these all on your own, you won’t achieve the glory of that French polymath, but you will have “re-discovered” some “elements of truth of analysis” …You can give yourself a pat on the back!

Cheers,

Nalin Pithwa.

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