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Is there a good reference on the approximation of sums using integrals ?

I am familiar with Riemann sums, and with the Euler-Maclaurin formula.

Still, many times when doing a probability calculation, I encounter a sum (actually the limit of a sum) which I cannot force into either of the above.

Here's an example: $$ \frac{1}{n} \sum_{j=-C \sqrt{n}}^{C \sqrt{n}} e^{-a \frac{j^2}{n} + b \frac{j}{\sqrt{n}}}$$ where $j$ runs over all the integers in the interval $[-C\sqrt{n},C\sqrt{n}]$.

Teddy
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    Please, add some examples – Yuriy S Jul 28 '19 at 09:45
  • This question and the answers provide another summation method for integrals, maybe it might be useful https://math.stackexchange.com/q/3265474/269624 – Yuriy S Jul 28 '19 at 09:47
  • I'm not sure, but I think with some tinkering your example could be turned into a Riemann sum. Is this a very typical example? Sums of the kind $\sum_j e^{-a j^2+b j}$ are related to Jacobi theta functions – Yuriy S Jul 28 '19 at 10:44

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