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?
$$ \sum \limits_{i=1}^{\infty} f(i) = \lim_{n \to \infty} \sum \limits_{i=1}^{n} f(i) $$

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.
