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Euge
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Let ##c## be a complex number with ##|c| \neq 1##. Find $$\lim_{n\to \infty} \frac{1}{n}\sum_{\ell = 1}^n \frac{\sin(e^{2\pi i \ell/n})}{1-ce^{-2\pi i \ell/n}}$$
The limit of complex sums is a mathematical concept that refers to the value that a sequence of complex numbers approaches as the number of terms in the sequence increases towards infinity.
The limit of complex sums is typically calculated by taking the sum of all the terms in the sequence and then dividing by the number of terms in the sequence. This process is repeated as the number of terms in the sequence increases towards infinity.
The limit of complex sums is an important concept in mathematics as it allows us to understand the behavior of complex sequences and determine their convergence or divergence. It is also used in various mathematical applications such as in calculus, series, and differential equations.
The main difference between the limit of complex sums and the limit of real sums is that complex sums involve complex numbers, which have both a real and imaginary component. This means that the limit of complex sums can have a real and imaginary part, while the limit of real sums can only have a real part.
Yes, the limit of complex sums can have a complex value if the sequence of complex numbers being summed does not have a real limit. This means that the sequence is oscillating or spiraling towards a complex value as the number of terms increases towards infinity.