Number of Terms for Harmonic Series to Reach a Sum of 100

In summary, Julian Havil's book "Gamma-Exploring Euler's Constant" discusses the harmonic series and its properties, including its slow divergence towards infinity. However, one paragraph on page 23 mentions a calculation by John W. Wrench Jr, who determined the exact minimum number of terms needed for the series to sum past 100. This number is 15,092,688,622,113,788,323,693,563,264,538,101,449,859,497 and was not obtained by actually adding up the terms, but through precise approximations. While there are formulas to approximate the harmonic sum, they are not as accurate as Wrench's calculation. The accuracy of this calculation has been questioned,
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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.
 
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https://www.jstor.org/stable/2316476?seq=1#metadata_info_tab_contents
The American Mathematical Monthly, Vol. 78, No. 8 (Oct., 1971), pp. 864-870

He used really good approximations, so good that the error has to be below 1 additional term.
 
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I am still skeptical of such accuracy. Why not have super computer do a test. How many terms in an harmonic series do you need to exceed the sum of 27. This will be more manageable and might take a few hours of computer time. This harmonic series should have about a trillion terms to add up.
 
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  • #4
You can test it for smaller numbers with your own computer. Testing a few more cases with a supercomputer is not a good use of their time. A sound mathematical proof, checked by others, is better than a test for the first few numbers.
 
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FAQ: Number of Terms for Harmonic Series to Reach a Sum of 100

1. How many terms are required for the harmonic series to reach a sum of 100?

The harmonic series is an infinite series that diverges, meaning it does not have a finite sum. Therefore, it is not possible to reach a sum of 100 with a finite number of terms.

2. Is there a formula for calculating the number of terms needed for the harmonic series to reach a specific sum?

There is no known formula for calculating the number of terms needed for the harmonic series to reach a specific sum. However, there are approximations and bounds that can be used to estimate the number of terms.

3. What is the rate of convergence for the harmonic series?

The harmonic series has a slow rate of convergence, meaning it takes a large number of terms to reach a significant sum. It is estimated that it takes around 10 million terms to reach a sum of 20.

4. Can the harmonic series ever reach a sum of 100?

No, the harmonic series is a divergent series and does not have a finite sum. It continues to increase without bound, so it is not possible for it to reach a sum of 100.

5. Are there any practical applications for knowing the number of terms needed for the harmonic series to reach a certain sum?

The harmonic series has many applications in mathematics and physics, but the number of terms needed to reach a specific sum is not typically a useful or practical concept. It is more commonly used in theoretical discussions and as an example of a divergent series.

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