summation - Summing series in the form: $sum_{n=infty}^{infty}f(n)$ via Complex Methods?

$(1.)$

$$\sum_{n=\infty}^{\infty}f(n)$$

$(2.)$

$$\sum_{n=1}^{\infty}f(n)=\lim_{n\to\infty}\sum_{n=1}^{n}f(n)$$

How would via the tools of Complex Analysis approach the summation of series in the from defined in $(1.)$ via the tools of Complex Variables ? If possible provide applicable examples.

$$EDIT$$

Adding to our original question how would handle the upper bound and lower bound of sum defined in $(1.)$ as $n \, \rightarrow \infty$, how would generalize the approach seen in real-variable methods as seen and defined in $(2.)$.

Answer

I don't know what $f(n)$ you are interested in. There is Cauchy's Residue Theorem, Fourier Series with Complex Exponentials, Parseval's (Plancherel's) Identity, and Poisson Summation Formula. Have a look at the answers to proving $\zeta(2)=\frac{\pi^2}{6}$ in each of the links below: Complex Analysis Solution to the Basel Problem ($\sum_{k=1}^\infty \frac{1}{k^2}$) for the Residue Theorem, http://math.cmu.edu/~bwsulliv/basel-problem.pdf for Fourier Series and Parseval's Identity, http://www.libragold.com/blog/2014/12/poisson-summation-formula-and-basel-problem/ for Poisson Summation.