700 years of secrets of the Sum of Sums (paradoxical harmonic series)
Mathologer・32 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.
Related videos
Summary
00:00
"Harmonic Series: Paradoxical Properties and Discoveries"
- The harmonic series is a significant infinite mathematical object with paradoxical properties.
- Many calculus professors are unaware of the series' most remarkable paradoxical properties.
- Recent discoveries have unveiled more counter-intuitive facts about the harmonic series.
- The video consists of six chapters, each highlighting different aspects of the series.
- A thought experiment involving weights on a balance demonstrates the concept of balancing points.
- The balancing point for a two-kilo weight is one-third from the right.
- Stacking blocks to create overhangs on a cliff edge is explored, with the maximal overhang increasing with more blocks.
- The optimal stacking arrangement for maximal overhang with 20 blocks was discovered only a decade ago.
- The harmonic series sums to infinity, as proven by the French bishop Nicole Oresme almost 700 years ago.
- The series' slow divergence to infinity is showcased through the addition of a million terms.
16:03
"Harmonic Series: Calculating Partial Sums Efficiently"
- In 1968, mathematician John W. Wrench Jr. calculated that approximately 10^43 terms are needed to reach a partial sum of 100.
- The method of adding 10^43 terms individually is impractical due to the immense time it would take, even for powerful computers.
- The search for a formula to calculate the partial sum more efficiently leads to analyzing the pattern of odd numerators and even denominators in fractions.
- Odd over even fractions cannot result in integers, unlike other types of fractions, which can be integers in disguise.
- It is proven that all partial sums of the harmonic series follow the odd over even fraction pattern, except for the initial sum of 1.
- The partial sums of the harmonic series approach infinity without ever reaching an integer, passing each integer by smaller increments.
- The area under the curve of the function 1/x provides a visual representation of the harmonic series, aiding in understanding the partial sums.
- The natural logarithm of n plus one serves as a good approximation for the partial sum of the harmonic series, with gamma enhancing the accuracy.
- The formula incorporating gamma provides increasingly accurate approximations for larger partial sums, improving with higher values of n.
- The formula can be used to calculate the number of terms needed to exceed a sum of 100 and determine the overhang of a tower of blocks, showcasing the slow logarithmic growth compared to other tower structures.
30:56
"Summing Numbers with Harmonic Series Sub-series"
- 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, like gamma, a greedy algorithm can be used.
- The first term of the sub-series should be the largest harmonic reciprocal less than the target number.
- Subsequent terms are chosen to add up to the target number without exceeding it.
- Sub-series can either converge to a finite sum or diverge to infinity, like the sum of reciprocals of even or odd numbers.
- The sum of reciprocals of primes is infinite, despite the sparsity of primes.
- Kempner's series, excluding numbers with the digit nine, surprisingly has a finite sum.
- Robert Baillie discovered the sum of the no 9 series to be approximately 22.92, after Kempner's 1914 proof.
- Baillie also calculated good approximations for other digit-excluding series, like no zeros or no nines, with slow convergence rates.
