How to sum numerator and denominator of a fraction?

Question

I want to do sum over this. Can apply the summation to top and bottom separately?

$$\sum\limits_{i=1}^{n} \frac{-a(x_i-\mu)^2}{x_i}$$ $$=\frac{\sum\limits_{i=1}^{n}-a(x_i-\mu)^2}{\sum\limits_{i=1}^{n}x_i}$$

Is this correct?

Where can I find the rules to summations of a division, product, or addition?

Edit:
I want to solve for y, $$\frac{5}{y}=\sum\limits_{i=1}^{n} \frac{-a(x_i-\mu)^2}{x_i}$$ $$\frac{y}{5}=\sum\limits_{i=1}^{n} \frac{x_i}{-a(x_i-\mu)^2}$$

Is this correct?

Answer 1

No. For example $ \frac{3 + 2}{1 + 1} = \frac{5}{2} \neq 5 = \frac{3}{1} + \frac{2}{1} $. There are no basic rules for division or product. Of cource, $ \sum a_n + b_n = \sum a_n + \sum b_n $ which follows from definition.

One trick to use with products though is that $ \ln \prod a_n = \sum \ln a_n $.

Answer 2

One thing you can do which might be helpful is, assuming $x_i>0$ for all $i$, $$\frac{-a(x_i-\mu)^2}{x_i}=\frac{-a(x_i-\mu)^2}{(\sqrt{x_i})^2}=-a\left(\sqrt{x_i}-\frac{\mu}{\sqrt{x_i}}\right)^2$$

Answer 3

As the example in the comment shows nicely, no - Another simple example shows why this can't work: $$\frac 12 + \frac 12 \neq \frac 24 = \frac 12$$

Next: You might want to visit Wikipedia's entry for Summation, which includes a nice list of valid manipulations that hold for sums.