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

[Thecla]

TL;DR

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.

 

[mfb]

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.

 

[Thecla]

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.

 

[mfb]

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.