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I want to calculate the sum:

$$\sum_{k=1}^{1000} \frac{\lambda_1^k\exp(-\lambda_1)}{k!}(1-\sum_{n=0}^{k-1} \frac{1}{n!}exp(-\lambda_2 x)(\lambda_2 x)^n)$$

Wolfram Alpha seems to be having trouble with the second summation, I think the problem is that the stopping point of the second summation depends on the first summation. Is Wolfram Alpha able to solve this summation or is there different software I can use?

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  • $\begingroup$ In the second sum, factor out $\exp(-\lambda)$. The result doesn't have a particularly nice form and is known as the exponential sum function. $\endgroup$ – Simply Beautiful Art Dec 4 '17 at 15:19
  • $\begingroup$ Ok, but that form still has summations dependant on a variable. Unless Wolfram Alpha has a function for the gamma function. $\endgroup$ – John Meighan Dec 4 '17 at 15:23
  • $\begingroup$ My point was that I doubt there is a good closed form unless maybe rearranging the order of the sums does something. $\endgroup$ – Simply Beautiful Art Dec 4 '17 at 15:27
  • $\begingroup$ Well Wolfram Alpha doesn't really need a good closed form to do calculations? $\endgroup$ – John Meighan Dec 4 '17 at 15:29
  • $\begingroup$ WA can't do everything mate, and if the input/calculations get too complicated it'll quit on you. $\endgroup$ – Simply Beautiful Art Dec 4 '17 at 15:31

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