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Thecla

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- TL;DR Summary
- The number of terms for the harmonic series to reach a sum of 100 is very large. How did a mathematician calculate that number?

I am reading an interesting book by Julian Havil called:" Gamma-Exploring Euler's Constant."

Much of the book is devoted to the harmonic series,a slowly diverging series that tends toward infinity.However,one paragraph puzzles me. On p. 23 he says:

" In 1968 John W. Wrench Jr calculated the exact minimum number of terms for the series to sum past 100; that number is 15 092 688 622 113 788 323 693 563 264 538 101 449 859 497. Certainly he did not add up the terms. Imagine a computer doing so and suppose that it takes it a billionth of a second to add each new term to the sum and that we set it adding and let it continue doing so indefinitely. The job will have been completed in not less than 3.5X 10^17(American) billion years."That is it. Then he goes on to the next topic.How did John W Wrench Jr. calculate the exact minimum number of terms needed to exceed 100 with such precision. Julian Havil doesn't tell us. I know there are formulas to approximate the harmonic sum, but they are approximate.

Much of the book is devoted to the harmonic series,a slowly diverging series that tends toward infinity.However,one paragraph puzzles me. On p. 23 he says:

" In 1968 John W. Wrench Jr calculated the exact minimum number of terms for the series to sum past 100; that number is 15 092 688 622 113 788 323 693 563 264 538 101 449 859 497. Certainly he did not add up the terms. Imagine a computer doing so and suppose that it takes it a billionth of a second to add each new term to the sum and that we set it adding and let it continue doing so indefinitely. The job will have been completed in not less than 3.5X 10^17(American) billion years."That is it. Then he goes on to the next topic.How did John W Wrench Jr. calculate the exact minimum number of terms needed to exceed 100 with such precision. Julian Havil doesn't tell us. I know there are formulas to approximate the harmonic sum, but they are approximate.