TITLE: When does the summation of a quotient equal the quotient of summations?
QUESTION [7 upvotes]: That is, under what conditions would 
$$
\sum_{i = 1}^n \frac{a_i}{b_i}= \frac{\sum_{i = 1}^n a_i}{\sum_{i = 1}^n b_i} 
$$
be true?  What about for infinite summations, i.e. when $n \rightarrow \infty$?

REPLY [5 votes]: The following result might be of help:
Theorem: If $a_i\ge 0$, $b_i>0$ for all $i$ and not all $a_i$s are zero, then
$$
\sum_{i = 1}^n \frac{a_i}{b_i}= \frac{\sum_{i = 1}^n a_i}{\sum_{i = 1}^n b_i} 
$$
does not hold.
Proof: If $a_i\ge 0$, $b_i>0$ for all $i$, we can show by mathematical induction that
$$\frac{\sum_{i = 1}^n a_i}{\sum_{i = 1}^n b_i}\le
\max_{1\le i\le n}\frac{a_i}{b_i}.\ \ (*)$$
and
$$\max_{1\le i\le n}\frac{a_i}{b_i}\le\sum_{i = 1}^n \frac{a_i}{b_i}.\ \ (**)$$
The equality of $(*)$ holds when $a_1/b_1=\cdots=a_n/b_n$, and
the equality of $(**)$ holds when at most one of $a_i$s are nonzero;
this suggests the equality of $(*)$ and $(**)$ hold at the same time only when all $a_i$s are zero.
Therefore, if $a_i\ge 0$, $b_i>0$ for all $i$ and not all $a_i$s are zero,
$$
\sum_{i = 1}^n \frac{a_i}{b_i}>\frac{\sum_{i = 1}^n a_i}{\sum_{i = 1}^n b_i}.
$$