tworitdash
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From my physical problem, I ended up having a sum that looks like the following.
S_N(\omega) = \sum_{q = 1}^{N-1} \left(1 - \frac{q}{N}\right) \exp{\left(-\frac{q^2\sigma^2}{2}\right)} \cos{\left(\left(\mu - \omega\right)q\right)}
I want to know what is the sum when N \to \infty. Here, \omega is where this is computed and \mu and \sigma are constants. Can this be reduced to an expression (a function of variables \omega, \mu and \sigma) ?
I proceeded with trying to show that it is indeed convergent. S_N(\omega) - S_{N - 1}(\omega) = (1 - \frac{N-1}{N}) \exp{\left(-\frac{(N-1)^2\sigma^2}{2}\right)} \cos{\left(\left(\mu - \omega\right)(N-1)\right)} + \sum_{q = 1}^{N-2} q(\frac{1}{N-1} - \frac{1}{N}) \exp{\left(-\frac{q^2\sigma^2}{2}\right)} \cos{\left(\left(\mu - \omega\right)q\right)}
This difference goes to 0 when N \to \infty.
S_N(\omega) = \sum_{q = 1}^{N-1} \left(1 - \frac{q}{N}\right) \exp{\left(-\frac{q^2\sigma^2}{2}\right)} \cos{\left(\left(\mu - \omega\right)q\right)}
I want to know what is the sum when N \to \infty. Here, \omega is where this is computed and \mu and \sigma are constants. Can this be reduced to an expression (a function of variables \omega, \mu and \sigma) ?
I proceeded with trying to show that it is indeed convergent. S_N(\omega) - S_{N - 1}(\omega) = (1 - \frac{N-1}{N}) \exp{\left(-\frac{(N-1)^2\sigma^2}{2}\right)} \cos{\left(\left(\mu - \omega\right)(N-1)\right)} + \sum_{q = 1}^{N-2} q(\frac{1}{N-1} - \frac{1}{N}) \exp{\left(-\frac{q^2\sigma^2}{2}\right)} \cos{\left(\left(\mu - \omega\right)q\right)}
This difference goes to 0 when N \to \infty.
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