Sum of square roots vs. square root of sum in summation notation

Question

Recently, a TA of mine wrote

$$ \sum_{i=0}^n \sqrt x_i $$

to represent the sum

$$ \sqrt {x_0 + x_1 + \cdots + x_{n-1} + x_n}. $$

Is this an incorrect use of summation notation? My instinct is that the above sum is better represented by the following:

$$ \sqrt {\sum_{i=0}^n x_i} $$

Which is correct?

Answer 1

Pay attention $$ \sum_{i=0}^n \sqrt x_i=\sqrt {x_0} + \sqrt {x_1} + \cdots + \sqrt {x_n} $$ But $$ \sqrt {\sum_{i=0}^n x_i}=\sqrt {x_0 + x_1 + \cdots + x_n} $$ which is different.