diff --git a/.ipynb_checkpoints/Function_Behavior_Cheatsheet-checkpoint.ipynb b/.ipynb_checkpoints/Function_Behavior_Cheatsheet-checkpoint.ipynb index 398ee83..279a34c 100644 --- a/.ipynb_checkpoints/Function_Behavior_Cheatsheet-checkpoint.ipynb +++ b/.ipynb_checkpoints/Function_Behavior_Cheatsheet-checkpoint.ipynb @@ -24,7 +24,15 @@ "source": [ "## Factorials\n", "\n", - "Factorials are exceeded in growth rate by _[double exponential functions](https://en.wikipedia.org/wiki/Double_exponential_function)_ (e.g. $f(x) = e^{c^x}$) only.\n", + "*\n", + "Factorials are exceeded in growth rate only by _[double exponential functions](https://en.wikipedia.org/wiki/Double_exponential_function)_ (e.g. $a_n = n^n \\space \\text{or} \\space b_n = e^{x^c}$).\n", + "\n", + "$$\n", + "\\text{For instance, if} \\space \\{a_n\\} = \\frac{n!}{n^n} \\space \\text{, then} \\space\n", + "\\lim_{n\\to\\infty} \\{a_n\\} = 0\n", + "$$\n", + "\n", + "As $n\\to\\infty \\text{,}$ $\\{a_n\\} \\to 0$ because $n^n$ grows faster than $n!$.\n", "\n", "More details about growth rates and other data about factorials can be found on [Wikipedia](https://en.wikipedia.org/wiki/Factorial#Rate_of_growth_and_approximations_for_large_n).\n", "\n", diff --git a/Function_Behavior_Cheatsheet.ipynb b/Function_Behavior_Cheatsheet.ipynb index 398ee83..279a34c 100644 --- a/Function_Behavior_Cheatsheet.ipynb +++ b/Function_Behavior_Cheatsheet.ipynb @@ -24,7 +24,15 @@ "source": [ "## Factorials\n", "\n", - "Factorials are exceeded in growth rate by _[double exponential functions](https://en.wikipedia.org/wiki/Double_exponential_function)_ (e.g. $f(x) = e^{c^x}$) only.\n", + "*\n", + "Factorials are exceeded in growth rate only by _[double exponential functions](https://en.wikipedia.org/wiki/Double_exponential_function)_ (e.g. $a_n = n^n \\space \\text{or} \\space b_n = e^{x^c}$).\n", + "\n", + "$$\n", + "\\text{For instance, if} \\space \\{a_n\\} = \\frac{n!}{n^n} \\space \\text{, then} \\space\n", + "\\lim_{n\\to\\infty} \\{a_n\\} = 0\n", + "$$\n", + "\n", + "As $n\\to\\infty \\text{,}$ $\\{a_n\\} \\to 0$ because $n^n$ grows faster than $n!$.\n", "\n", "More details about growth rates and other data about factorials can be found on [Wikipedia](https://en.wikipedia.org/wiki/Factorial#Rate_of_growth_and_approximations_for_large_n).\n", "\n",