0

Consider the sum $S=\sum_{x\in I}P(x)$, where $P(x)$ are positive real numbers. When the index set $I$ is finite, $S$ is of course finite. When $I$ is countably infinite, it is also possible that $S$ is finite. For example $\sum_{x\in \mathbb{N}}\frac{1}{2^x}$. Is it true that $S$ cannot be finite if $I$ is uncountable?

velut luna
  • 9,961
  • 4
    How do you define the sum in that case? Note that even the countable case is not really a sum any longer, but a limit of partial sums. – Tobias Kildetoft Mar 11 '15 at 13:05
  • 1
    http://math.stackexchange.com/questions/20661/the-sum-of-an-uncountable-number-of-positive-numbers – Hasan Saad Mar 11 '15 at 13:14
  • Unless $P(x)$ is zero for all but countably many terms, then I know of no good way to make sense of such a sum (over an uncountable index set). – TravisJ Mar 11 '15 at 13:14

0 Answers0