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Let $(a_n)$ be the Taylor development of the exponential function as follows: \begin{eqnarray} a_n(x) = \sum^{n}_{k = 0} \frac{x^k}{k!} \end{eqnarray}

Let $(y_n)$ be a positive and strictly monotone sequence with limit $+\infty$.

Question Can we prove the following limit ?

\begin{eqnarray} \lim_{n \to +\infty } \frac{ a_n(y_n) }{e^{y_n}} = 1 \end{eqnarray}

If not possible, can we prove that at least this limit $\lim_{n \to +\infty } \frac{ a_n(y_n) }{e^{y_n}} > 0$ ?

You can find a French version of this question here: Calculer la limite de la suite suivante: $\frac{ a_n(y_n) }{e^{y_n}}$?

Mostafa Ayaz
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InfiniteMath
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    We have $$\frac{a_n(n^3)}{e^{n^3}} \le \frac{1+n^2+\cdots+\frac{n^{3n}}{n!}}{1+n^2+\cdots+\frac{n^{3(n+1)}}{(n+1)!}}\le\frac{(n+1)\frac{n^{3n}}{n!}}{\frac{n^{3(n+1)}}{(n+1)!}}\to 0.$$ – Riemann Feb 20 '23 at 00:25
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    @InfiniteMath I highly recomend you to take a look at this post that proves that, when $y_n=n$, the limit is $1/2$. – K. Makabre Feb 20 '23 at 00:36
  • Thank you Danka very much for your input. Really wonderful! – InfiniteMath Feb 20 '23 at 02:17
  • There is post that proves something quite similar: https://math.stackexchange.com/questions/160248/evaluating-lim-limits-n-to-infty-e-n-sum-limits-k-0n-fracnkk?noredirect=1&lq=1 – InfiniteMath Feb 20 '23 at 23:10
  • There is also this post that proves another similar result: https://math.stackexchange.com/questions/3339131/asymptotics-of-partial-exponential-sum-sum-limits-k-0a-n-fracnkk?noredirect=1&lq=1 – InfiniteMath Feb 20 '23 at 23:11
  • Another good idea here: https://math.stackexchange.com/questions/3764175/lim-limits-n-rightarrow-infty-e-2n-sum-k-0n-frac2nkk?rq=1 – InfiniteMath Feb 20 '23 at 23:15
  • And here: https://math.stackexchange.com/questions/1166061/the-limit-lim-n-to-infty-fract-nnen-where-t-nx-is-the-taylor-po – InfiniteMath Feb 20 '23 at 23:15

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