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$(1.)$

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

$(2.)$

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

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.

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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 $(2.)$.

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  • $\begingroup$ Is there a particular $f(n)$ you're interested in? $\endgroup$ – Larry B. May 16 '17 at 17:35
  • $\begingroup$ $f(n)$ can be any function really I just what to see some approach's and tools from complex variables being used on infinite sums. $\endgroup$ – Zophikel May 16 '17 at 17:37
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    $\begingroup$ Shouldn't the lower limit of the sum be $-\infty$ ? $\endgroup$ – Vivek Kaushik May 16 '17 at 17:49
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    $\begingroup$ @VivekKaushik I don't see how that is a justified assumption. Perhaps the asker is referring to the limit as the upper and lower bound both approach infinity at the same rate? I could see that being a viable and intriguing question. $\endgroup$ – The Great Duck May 16 '17 at 18:45
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    $\begingroup$ A common method is to use that the function $g(z) = \pi\cot(\pi z)$ has residue $1$ at the integers so the integral $\oint f(z)g(z){\rm d}z$ over some appropriate contour containing all the integers will via the residue theorem give rise to $\sum f(n)$ plus additional terms coming from the poles of $f(z)$. $\endgroup$ – Winther May 17 '17 at 1:11
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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.

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