Is it true that $ \sum \limits_{i=1}^{\infty} f(i) = \lim_{n \to \infty} \sum \limits_{i=1}^{n} f(i) $? Is the equality below true?
$$ \sum \limits_{i=1}^{\infty} f(i) = \lim_{n \to \infty} \sum \limits_{i=1}^{n} f(i) $$
 A: According to the modern definition, the meaning (by definition) of $\sum_{i=1}^\infty f(i) $ is that it is synonymous with the limit $\lim_{n\to \infty }\sum_{i=1}^n f(i)$. So, the short answer is that the two expressions exists or do not exist together, and when they exist they are equal. Though this is a completely vacuous thing to discuss since it is true by definition, that is, there is nothing to prove here. 
Historically, it took quite some time for both the limit notion and the infinite sum notion to converge to the modern definition we use today. Many prominent mathematicians (Euler, Bolzano, Cauchy and others) introduced various ways to interpret the meaning of infinite sum. Some of these definitions are contradictory and for some of those definitions the two expressions you wrote are not the same. Quite famously is the case of the series $1-1+1-1+1-1+\cdots $ which (by erroneously evaluating the equation $\frac{1}{1-x}=1+x+x^2+\cdots $, valid for $|x|<1$ at $x=-1$) was claimed by some to be equal to $\frac{1}{2}$. This is not just a silly mistake. It stems from deep confusion and disagreement about what function should mean back in the early days of calculus. 
A: Since it wasn't specified, I'm assuming that the codomain of $f$ is something in which the symbols used make sense.
The equality $\displaystyle  \sum \limits_{i=1}^{\infty} f(i) = \lim_{n \to \infty} \sum_{i=1}^{n} f(i)$ holds by definition, if $\displaystyle \lim \limits_{n \to \infty} \sum \limits_{i=1}^{n} f(i)$ exists.
If the limit doesn't exist, then the LHS is devoided of meaning and the 'equality' is neither true, nor false, it simply doesn't make sense.
