Euler on infinite series
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[edit] Divergent series
Leonhard Euler succinctly described a potential foundation for his treatment of divergent series in a calculus textbook published in 1755[1]:
| “ | Let us say, therefore, that the sum of any infinite series is the finite expression, by the expansion of which the series is generated. In this sense the sum of the infinite series 1 − x + x2 − x3 + · · · will be 1⁄1+x, because the series arises from the expansion of the fraction, whatever number is put in place of x. If this is agreed, the new definition of the word sum coincides with the ordinary meaning when a series converges; and since divergent series have no sum, in the proper sense of the word, no inconvenience can arise from this new terminology. Finally, by means of this definition, we can preserve the utility of divergent series and defend their use from all objections. | ” |
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—Euler, 1755 |
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See also the history of Grandi's series.
[edit] Notes
- ^ Euler (1755), Part 1, Chapter 3, #111, pp.78-79; English translation by Bromwich (p.322). Kline's translation (1983, p.313) is almost identical.
[edit] References
- Bromwich, T.J. [1908] (1926). An Introduction to the Theory of Infinite Series, 2e.
- Euler, Leonhard (1755). Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum.
- Kline, Morris (November 1983). "Euler and Infinite Series". Mathematics Magazine 56 (5): 307-314.
