Definition: Let $F_n(s) = \frac{1}{s^{n+1}(1+s)^n}$ be the Laplace transform of $f_n(t)$.

Required Result: To show $\lim_{n\rightarrow\infty}f_n(n+n/e) < o(n)$.

Ideas: Let $G_n(s)=\frac{1}{s^{n+1}}$. Using standard Laplace transform properties:

  1. $g_n(t)= \frac{t^{n}}{n!}$. Using Stirling's approximation, $\lim_{n\rightarrow\infty}g(n/e) = 0$.
  2. $h_n(t) = g_n(t-n)$. We have $H_n(s) = \frac{1}{s^{n+1}e^{sn}}$ and $\lim_{n\rightarrow\infty}h_n(n+n/e) = 0$.

Note that $F_n(s)$ and $H_n(s)$ 'look' similar. The $(1+s)$ term corresponds to damping whereas $e^s$ term corresponds to delay. Can we use this idea to prove the result?

Any ideas/references would be appreciated!


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