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I am interested in analysis of the partial MacLaurin sum for the exponential decay function:

$$E_n(\lambda) \equiv \sum_{i=0}^n \frac{(-\lambda)^i}{i!} \quad \quad \quad \text{for } \lambda \geqslant 0 \text{ and } n \geqslant 0.$$

Obviously this function has the limit $E_n(\lambda) \rightarrow \exp(-\lambda)$ as $n \rightarrow \infty$, which is the exponential decay function. For the purposes of some probability work, I am interested in finding out conditions on $\lambda$ that ensure that this function is non-negative.

It seems to me that large values of $\lambda$ might lead this partial sum to be negative sometimes, so I am wondering if there is any (non-trivial) upper bound I can impose on $\lambda$ to ensure that the function is non-negative. I have in mind some upper bound (that can depend on $n$) where the condition $0 \leqslant \lambda \leqslant U(n)$ is sufficient to ensure that $E_n(\lambda) \geqslant 0$ for all $n=0,1,2,...$.

Question: Is there a (non-trivial) sufficient condition I can impose on $\lambda$ (either an upper-bound depending on $n$ or some other condition) that will ensure that the function above is non-negative?

Ben
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  • You haven't defined "nontrivial", so I don't know if this is helpful: when $0\le\lambda\le1$, the summands are decreasing in absolute value, and so the partial sums are nonnegative by the reasoning behind the alternating series test. Emperically, the even-$n$ partial sums are always nonnegative, while the odd-$n$ partial sums are nonnegative for $\lambda\le U(n)$ where $U(n)$ seems quite linear with slope approximately $0.28$. – Greg Martin Jul 08 '21 at 05:14
  • That is helpful, but I suppose I am interested getting a form for the upper bound $U(n)$ in cases that go beyond the simple case of $n=1$. Finding the bounds seems to involve finding high-order polynomial roots, which does not give a closed form, so I am hoping there is some non-trivial closed form that is a sufficient bound (but not necessarily the best possible bound). – Ben Jul 08 '21 at 07:11
  • You could take a more complicated problem, the location of the roots, to get an idea for your problem. See https://math.stackexchange.com/questions/109360/roots-of-the-incomplete-gamma-function, https://math.stackexchange.com/questions/131479/complex-zeros-of-the-polynomials, and the "Linked" lists there. – Lutz Lehmann Jul 08 '21 at 07:26

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