Thank you very much for the response. How did you arrive at this answer? Did you use the Bessel function expansion of the term $e^{i \tau (c_1 - c_2 e^{-c_3 x})} $ ?

@SidharthGhoshal no, they are not integers.

I want to construct a covariance function of a signal that has a skewed Gaussian like power spectrum. The covariance function is nothing but the inverse Fourier of the power spectrum. That is why I wanted this. A closed form would help because of computational issues.

I am adding a new purpose based on approximations.

I also want to understand why $\epsilon_{n}$ can only take countably many values. I am not a mathematician, so my understanding can be really basic. From your approach, all I can understand is that probably the two terms involved in the original equation behave differently as $n$ increases so you introduced $\epsilon_{1n}$ and $\epsilon_{2n}$. Then, you dealt them separately to find convergence of each and then took a maximum? I think this is what I was looking for. When I said $\epsilon$, it is a common non mathematician way of saying convergence. I am happy I am learning new things here. :)

I want to understand why we use certain inequalities sometimes to find easier ways to show convergence. For example, when we say $\epsilon_{2n} = \frac{1}{n}\sum_{q=1}^{n-1} a_q q \leq \frac{1}{n}\sum_{q=1}^{\infty} a_q q$ . Logically it makes sense. However, I want to understand why it is used in this context. It could be any number less than $\infty$ and bigger than $n-1$. How do we acknowledge the error we are introducing by such inequalities.

Thank you very much indeed for the detailed answer. I will look into it line by line. When you said $n$ will do instead of $N$, do you mean only in terms of notation, or something else? And, when you said the question is stated poorly, do you mean the definition of the error stated poorly? I want to understand.

Thank you so much for the answer again. It gave me a lot of perspective. My simulations agree with this.