700 years of secrets of the Sum of Sums (paradoxical harmonic series)
Mathologer・2 minutes read
The harmonic series has paradoxical properties and recent discoveries have revealed more counter-intuitive facts about it, with the French bishop Nicole Oresme proving its sum to be infinity almost 700 years ago. Mathematicians have analyzed the series using visual representations and formulas to calculate partial sums more efficiently, demonstrating the slow logarithmic growth compared to other structures like towers of blocks.
Insights
- The harmonic series, despite its slow divergence to infinity, showcases paradoxical properties and has been a subject of recent discoveries uncovering counter-intuitive facts, with mathematicians like Nicole Oresme and John W. Wrench Jr. contributing to its understanding over centuries.
- Utilizing a formula involving gamma, the harmonic series can be approximated effectively, aiding in calculations for various applications such as determining the number of terms needed to exceed a specific sum or analyzing tower structures' overhang, highlighting its unique role in representing numbers through sub-series and the surprising convergence properties of digit-excluding series discovered by mathematicians like Robert Baillie.
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Recent questions
What is the harmonic series and its paradoxical properties?
The harmonic series is an infinite mathematical series that sums to infinity, as proven by Nicole Oresme almost 700 years ago. It showcases paradoxical properties, such as the slow divergence to infinity and the fact that all partial sums follow an odd over even fraction pattern, except for the initial sum of 1. This series has been a subject of recent discoveries uncovering more counter-intuitive facts, making it a significant object in mathematics.
How can the harmonic series be visually represented?
The harmonic series can be visually represented by the area under the curve of the function 1/x. This visual aid helps in understanding the partial sums of the series, as each partial sum approaches infinity without ever reaching an integer. The natural logarithm of n plus one serves as a good approximation for the partial sum, with gamma enhancing the accuracy. This representation aids in showcasing the slow logarithmic growth compared to other tower structures.
What is the optimal stacking arrangement for maximal overhang with blocks?
The optimal stacking arrangement for maximal overhang with blocks was discovered only a decade ago. Stacking blocks to create overhangs on a cliff edge showcases that the maximal overhang increases with more blocks. The method of stacking the blocks in a specific arrangement allows for achieving the greatest overhang possible, highlighting the intricate balance and mathematical principles involved in such structures.
How can the partial sum of the harmonic series be calculated efficiently?
The partial sum of the harmonic series can be calculated efficiently by analyzing the pattern of odd numerators and even denominators in fractions. By understanding that odd over even fractions cannot result in integers, a formula incorporating gamma can provide increasingly accurate approximations for larger partial sums. This method leads to a more efficient calculation of the number of terms needed to exceed a sum of 100, showcasing the slow logarithmic growth of the series.
Can every positive number be represented as a sum of sub-series of the harmonic series?
Yes, every positive number can be represented as a sum of infinitely many sub-series of the harmonic series. To find a sub-series summing to a specific number, a greedy algorithm can be used, where the first term should be the largest harmonic reciprocal less than the target number. Subsequent terms are then chosen to add up to the target number without exceeding it. These sub-series can either converge to a finite sum or diverge to infinity, showcasing the versatility and complexity of the harmonic series in representing various numbers.