今天是第一次正式写一篇关于自己的博客

因此有很多事项待处理，比如对于博客网页的改进，如何更好地使用 HTML 和 LaTeX 等等。
后天就到了博主的期末，愿正常发挥。

关于数列和级数的收敛性

Definition. A sequence is a function
\[ a : \mathbb{N}_{>0} \to \mathbb{R} \]

Definition.
\[ (a_n) \text{ converges to} \ (a \in \mathbb{R} ) \] iff

$$ \forall \varepsilon > 0, \exists N \in \mathbb{N} \text{ such that } \forall n \geq N, \ |a_n - a| < \varepsilon $$

We write this as $ \lim_{n \to \infty} a_n = a $

Otherwise，it is divergent.

收敛性检验

1.) ratio test： $$ \scriptsize \begin{aligned} \text{If a sequence } (a_n) \text{ satisfies that } \ \left| \frac{a_{n+1}}{a_n} \right| \to L < 1, \ \text{then } a_n \to 0. \end{aligned} $$

2.) Monotonic and Bounded: $$ \scriptsize (a_n) \text{ is monotonically increasing and bounded above, then } a_n \text{ converges to } $$ $ \scriptsize \\ a := \sup { a_i : i \in \mathbb{N} }, \text{written as } a_n \uparrow a. $

3.) Cauchy sequence

Definition.
A sequence ( (a_n) ) is a Cauchy sequence if and only if $$ \forall \varepsilon > 0, \exists N \in \mathbb{N} \text{ such that } \forall n, m \geq N, |a_n - a_m| < \varepsilon . $$

4.) Sandwish Test

$ \scriptsize \text{If for all} ( n \in \mathbb{N} ), we have ( c_n \leq a_n \leq b_n ), $ $ \scriptsize \text{and if the series} ( \sum c_n ) and ( \sum b_n ) \text{both converge,} $ $ \scriptsize \text{then the series } ( \sum a_n ) also converges. $

5.) conparison test

$\scriptsize If ( a_n b_n \to L \in \mathbb{R} ) as ( n \to \infty ), \text{and if the series } ( \sum b_n ) \text{ is absolutely convergent, } $

$ \scriptsize \text{then the series } ( \sum a_n ) \text{ is absolutely convergent.} $

6.)root-test

$$ \scriptsize \text{If a sequence } (a_n) \text{ satisfies that } \ a_n^{1/n} \to L < 1, \ \text{then } a_n \text{ is absulute converge}. $$

7.) alternating series test

$ a_n \geq 0 \text { and } a_n \text{ decreasing and } a_n \to 0 ( \text {or say } a_n \downarrow 0) \text{Then it is conditionally convergent} $

8.) p-test

$ \sum_{n=1}^{\infty} \frac{1}{n^{p}} \text{converges iff }p >1. $

后记 一旦忙活起了这个，就到了凌晨以二点，本来想做一些泰勒展开的东西，但是已经累的不想做了。第一次排版弄的稀碎，我应该学一学关于如何丰富博客功能的代码,比如超链接，插入和去除。我已经花了大半个小时才搞明白我能否用latex……。总之各位愉快。