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E710.pdf

710RevisedVersion2.pdf

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Summary

Euler considered a series s = 1 + ab/1 · c x + (a+1)(b+1)/2 · (c+1) x^2 + ... and found its sum seems impossible to exhibit in general, although it terminates and its sum is expressed in finite terms when either a or b is a negative number. He transformed the series into z = 1 + αβ/1 · c x + (α+1)(β+1)/2 · (c+1) x^2 + ..., where α = c - a and β = c - b, which terminates when either α or β is a positive integer. The transformation is achieved through a differential equation of second order, and Euler found that s = (1 - x)^(c-a-b) * z. He also derived the series for z through a direct method, assuming a power series for z and substituting it into the differential equation.

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Description

Euler considered a series s = 1 + ab/1 · c * x + (a+1)(b+1)/2 · (c+1) * x^2 + ...

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File Format: PDF
File Size: 136 KB
Pages: 13
Language: EN
Total Downloads: 31
Last Updated: 4 months ago

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