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Does $\sum_{a}^{b}x=-\sum_{b}^{a}x $ is correct?

violettagold
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1 Answers1

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No, reversed indexes are interpreted as either

  • no term in the sum (i.e. $0$) or

  • same as in the normal order.

But never opposite of the sum.

  • So I can say that $\sum_{a}^{b}x=\sum_{b}^{a}x $ – violettagold Mar 09 '20 at 08:36
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    @violettagold: as written, but that depends on the exact convention. –  Mar 09 '20 at 08:36
  • Can you please explain me what is exact convention? – violettagold Mar 09 '20 at 08:48
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    @violettagold: no, it is up to you. –  Mar 09 '20 at 08:48
  • For integrals over real intervals we do have the convention that $\int_b^a f(x)dx=-\int_a^b f(x)dx$ when $a < b$, so we could extend this to sums, which are a type of integral after all. – Henno Brandsma Mar 09 '20 at 08:54
  • @HennoBrandsma: for integrals, this is a quite natural convention by the Fundamental Theorem of Algebra (definite integral as the delta of the antiderivative). For sums, this is less intuitive as they are rarely obtained as "anti-differences). –  Mar 09 '20 at 09:27
  • True enough, but it probably is the reason for asking the question in the first place. – Henno Brandsma Mar 09 '20 at 09:28
  • @HennoBrandsma: For integrals, it's natural to change the sign, since “$dx$ is negative when you go backwards”. But for sums, there is no $dx$, so $a+b+c$ is the same as $c+b+a$ in the opposite order, not $-(c+b+a)$. – Hans Lundmark Mar 09 '20 at 09:32