Stirling's Formula and Its Simple Proof

In problem 1 (f) of tutorial note 10 we have mentioned that the power series  \[

\sum_{n=1}^\infty \frac{n!}{n^n}x^n

\] diverges when $|x|$ is the radius of convergence $e$. In fact, we can show that \[
n!\brac{\frac{e}{n}}^n\to \infty
\] by Stirling's Formula (in the simplest form), which is the following asymptotic result:

Theorem. We have $\dis n!\sim \sqrt{2\pi n} \brac{\frac{n}{e}}^n$ as $n\to \infty$.

Tutorial note 11

Tutorial note 10

Conditional and Absolute Convergence

In the discussion on convergence of series I forgot to mention the following two important concepts:

Definition.
(i) A series $\sum_{n=1}^\infty a_n$ is said to converge absolutely if $\sum_{n=1}^\infty |a_n|$ converges.
(ii) A series is said to converge conditionally if $\sum_{n=1}^\infty a_n$ converges but $\sum_{n=1}^\infty |a_n|$ diverges.

Tutorial note 9

Tutorial note 8

$\{b^n\}\ll \{n!\}\ll n^n$

More on Improper Integrals; $f\ge0$, $\int_a^\infty f\,dx<\infty\Rightarrow \lim_{x\to\infty}f(x)=0?$

In this course we shall mainly focus on improper integrals of nonnegative functions (i.e., a function $f$ such that $f(x)\ge 0$ for each $x$ in the domain). For nonnegative functions, \[\int_\square^\square f\,dx\text{ converges}\iff \int_\square^\square f\,dx <\infty,\] hence sometimes we conclude something converges by just writing $\int_\square^\square f\,dx <\infty$.

An important case (corollary 2 below) needs to be mentioned, since it is impossible to explain everything in tutorial, I decide to provide more detail in this blog (also typing maths in blogger is very easy!).

Tutorial note 7

Mathematical Induction

Detail can be seen here.

Tutorial note 6

Tutorial note 5

Tutorial note 4

Tutorial note 3