Pioneer Formula
Calculus 2 – Infinite Series Question?
The great Swiss mathematician Leonhard Euler sometimes reached incorrect conclusions in his pioneering work in infinite series. For example, Euler concluded that 1 / 2 = 1 – 1 + 1-1 + … and -1 = 1 + 2 + 4 + 8 + … substituting x =- 1 and x = 2 in the formula 1 / (1-x) = 1 + x + x ^ 2 + x ^ 3 + … What was the problem with your reasoning?
For both series, we see that Euler's equation uses the sum of infinite series a / (1-r) [a = first term, r = ratio between the terms], but that only works when ABS (r) is less than one, which is not the case here. [r =- 1 the first r = 2 in the second, so it varies the first and second series diverges).
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