In summary, the limit of a complex sum is the value that a sequence of complex numbers approaches as the number of terms in the sum approaches infinity. To find the limit, various techniques such as the squeeze theorem, direct evaluation, or the use of known limit properties can be used. Some common properties of limits of complex sums include linearity and the squeeze theorem. The limit of a complex sum can be infinite, which can occur when the terms in the sum are increasing without bound or when the sum involves complex numbers with large magnitudes. Limits of complex sums have various applications in real-world situations, such as modeling population growth and analyzing numerical methods in scientific and engineering applications.
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Euge
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Let ##c## be a complex number with ##|c| \neq 1##. Find $$\lim_{n\to \infty} \frac{1}{n}\sum_{\ell = 1}^n \frac{\sin(e^{2\pi i \ell/n})}{1-ce^{-2\pi i \ell/n}}$$
\begin{align*}
\sin (x+iy) & = \dfrac{e^{i x - y} - e^{-i x + y}}{2i}
\nonumber \\
& = \dfrac{e^{i x} - e^{-i x}}{2i} \dfrac{e^y + e^{- y}}{2} + i \dfrac{e^{i x} + e^{-i x}}{2} \dfrac{e^y - e^{- y}}{2}
\nonumber \\
& = \sin x \cosh y + i \cos x \sinh y
\nonumber \\
& =u (x,y) + i v (x,y)
\end{align*}
Note
\begin{align*}
u_x & = \cos x \cosh y ,
\nonumber \\
u_y & = \sin x \sinh y
\nonumber \\
v_x & = - \sin x \sinh y ,
\nonumber \\
v_y & = \cos x \cosh y .
\end{align*}