Abstract
If we add infinitely many numbers (more precisely, if we take the sum of an infinite sequence of numbers), then we get an infinite series.
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Notes
- 1.
This viewpoint does not treat an infinite sum as an expression whose value is already defined, but instead, the value is “gradually created” with our method. From a philosophical viewpoint, the infinitude of the series is viewed not as “actual infinity” but “potential infinity.”
- 2.
There are some who actually mean the series (s n ) when they write \( \sum _{n=1}^{\infty }a_{n} \). We do not follow this practice, since then, the expression \( \sum _{n=1}^{\infty }a_{n} = A \) would state equality between a sequence and a number, which is not a good idea.
- 3.
The name comes from the fact that the wavelengths of the overtones of a vibrating string of length h are \( h/n\ (n = 2, 3,\ldots ) \). The wavelengths \( h/2,\,h/3,\ldots,h/8 \) correspond to the octave, octave plus a fifth, second octave, second octave plus a major third, second octave plus a fifth, second octave plus a seventh, and the third octave, respectively. Thus the series \( \sum _{n=1}^{\infty }(h/n) \) contains the tone and its overtones, which are often called harmonics.
- 4.
Johann Bernoulli (1667–1748) Swiss mathematician, brother of Jacob Bernoulli.
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Laczkovich, M., Sós, V.T. (2015). Rudiments of Infinite Series. In: Real Analysis. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2766-1_7
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