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Just a quick question. Not sure if I have all the tools to solve this and I'm looking for a little direction.

I'm looking at the following summation

$$\sum_{i=1}^{n} (1 - x_{i})$$

I know that $0 < x_{i} < 1$ for all $i$.

Is there a way to tell when this summation will be greater than $1$? Clearly if $n = 1$ it won't be. But is there any other information we can get from it?

Thanks all

Tiny Tim
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    Not if we don't know anything more about $x_i$. If $x_i = \frac 1{2^i}$ then the sumation divergers (there are an infinite number of terms that are greater than $\frac 12$) to infinity. If $x_i = 1-\frac 1{2^i}$ then $1-x_i = \frac 1{2^i}$ and the sum is equal to $1$. We can have pretty much any possible result. – fleablood Oct 13 '22 at 03:57
  • You need to define $x_i$ – Claude Leibovici Oct 13 '22 at 06:17
  • It would seem that if $\sum_{j=1}^{n} x_{j} < n-1$ then the desired sum would be greater than $1$ otherwise more information is needed about $x_{i}$. – Leucippus Oct 13 '22 at 17:33

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