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COMMENTS
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"In 1968 John W. Wrench Jr calculated the exact minimum number of terms needed 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 10^-9 seconds 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 heen completed in not less than 3.5 * 10^17 (American) billion years." Havil.
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REFERENCES
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R. Baillie, Fun With Very Large Numbers, Arxiv preprint arXiv:1105.3943, 2011
Boas, R. P. and Wrench, J. W. "Partial Sums of the Harmonic Series." Amer. Math. Monthly 78, 864-870, 1971.
Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, page 23.
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