Oresme's genius was to make a new series which was…
““Oresme's genius was to make a new series which was definitely smaller than the harmonic series. He took the list of all unit fractions, and for any of them which did not have a power of two as a denominator, he replaced it with a smaller fraction which did. As all these new fractions were either the same or smaller, the total of this new series had therefore to be smaller than the sum of the harmonic series. But when Oresme grouped these fractions into runs, each of which added up to 1/2, he was left with a sum of an infinite sequence of 1/2s, which definitely diverges. This meant in turn that the greater harmonic series must also diverge. Oresme had proved that a sequence of ever-decreasing numbers could still be divergent. (His proof was lost for a while, and the same result was independently rediscovered in the 1600s.)””
About This Quote
This interpretation was drafted with AI assistance. It is one reading of the quote, not the author's own explanation.
Oresme showed that even a series of decreasing terms can diverge by grouping fractions into infinite halves, proving the harmonic series diverges.
In simple terms: Decreasing series can still diverge.
Key Takeaway
Check assumptions about convergence.
Themes
math
history
logic
Mood
analytical
inquisitive
Type
educational
historical
When to use this quote
- teaching
- research
- proof writing
- curriculum design
Key Concepts
convergence
divergence
series
grouping
Questions to Reflect On
- Can you think of other decreasing series that converge?
- How does grouping affect convergence?
A Different Perspective
The argument relies on specific grouping; other arrangements may behave differently.