Professor Stewart's Casebook of Mathematical Mysteries by Ian Stewart

Author:Ian Stewart [Stewart, Ian]
Language: eng
Format: epub
Tags: Mathematics, General
ISBN: 9781847654328
Google: HvFYBQAAQBAJ
Amazon: 0465054978
Publisher: Profile Books
Published: 2014-10-01T16:00:00+00:00

Random Harmonic Series

The infinite series

is called the harmonic series by mathematicians. The name is loosely related to music, where the overtones of a vibrating string have wavelengths 1/2, 1/3, 1/4, and so on, of the string’s fundamental wavelength. However, the series has no musical significance. It is known to be divergent, meaning that the sum up to n terms becomes as large as we wish if n is large enough. It diverges very slowly, but it does diverge. In fact, the first 2n terms add up to more than 1 + n/2. On the other hand, if we change the sign of every other term, we get the alternating harmonic series

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