Harmonic Series: Proving it diverges via an ingenious method

In this video, I show that the harmonic series, 1 + 1/2 + 1/3 + 1/4 + ..., diverges by showing its even power partial sums are greater than a similar but smaller series involving repeating 1/2 terms. This is a very clever derivation, and it was due to the French scholar Nicole Oresme from the 14th century.

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Timestamps

• Example 7: Show that the harmonic series is divergent – 0:00
• Solution: For this series, we will show that the even partial sums get larger – 0:26
• Writing each even partial sum as an inequality greater than a similar slightly smaller sum involving repeating terms of 1/2 – 2:01
• Pattern: The n-th partial sum is greater than 1 + n/2, which goes to infinity – 6:30
• The series of even power terms is divergent, therefore the harmonic series is divergent – 7:53
• This derivation is due to the French scholar Nicole Oresme (1323-1382) – 8:43

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