1. The formula in cell A4 is =B4+C6. On copying this formula to cell C6, the formula will change to ____________________
Answers
Answer:
In the usual sense, the series diverges to infinity. 1+2+4+8+⋯=∞ .
I’ll use the word “value” rather then the word “sum” when associating a number to a series since the sum of a series that diverges to infinity can only be ∞ .
The question of what value to assign to divergent series was a hot topic in the 1700s. In 1713 Leibniz wrote to Christian Wolff that the alternating series 1–1+1–1+1–1+⋯ should have the value 12 “based on the expansion of the fraction 11+1 .” The expansion is just long division. Likewise, he got 14 for the value of 1–2+3–4+⋯ by expanding 1(1+1)2 .
Euler looked at several divergent series, and wrote a paper De seriebus divergentibus (On divergent series) in 1746, published in 1760. An English translation with commentary by E.J. Barber and P.J. Leah is available at Historia Mathematica 3 (1976) pages 141–160, “Euler's 1760 paper on divergent series”. The series in the question was one of them. Here’s what he says about the series in question:
But those, who object to sums of divergent series, are judged to find their firmest support in the third type. For although the terms of these series continually increase and therefore it is possible for the terms to be gathered into a sum greater than an artitrarily given number, and this is the definition of infinite, yet the advocates for sums of such series are forced to admit that these sums are finite and indeed negative, that is less than zero. Namely, as the fraction 11−a yields upon division the series expansion 1+a+a2+a3+a4+⋯ , we ought to have
−1=1+2+4+8+16+⋯ , −12=1+3+9+27+81+⋯
and this is seen by opponents, not undeservedly, to be most absurd, since it is never possible to arrive at a negative sum through the addition of positive numbers. For this reason, they insist all the more on the necessity of adding the remainder mentioned above, as with this inserted it is clear that
−1=1+2+4+8+⋯+2n+2n+11−2
even if n is an infinite number.
Euler argues that a sum of positive numbers can be negative because numbers beyond beyond infinity may be considered negative. He notices that in the sequence …,14,13,12,11,10,1−1,1−21−31−4,… the numbers are increasing to infinity, and beyond that they are increasing negative numbers. This puts numbers in a circle and makes infinity a number. This is what is now called the Real projective line. In the image below, real numbers correspond to points on the circle except the top point which corresponds to ∞ which is identified with −∞ .
Image source: Chris Tees’ blog, Nested Tori
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