• Sequence and Series
• sequence $a_n$
  • sequence convergence
    • prove sequence convergence by function
    • sequence convergence preserve by chaining continuous function
    • convergence of $r^n$
  • sequence monotonicity
  • bounded sequence
    • monotonic sequence theorem
• series (infinite series)
  • $n$th partial sum $s_n$
  • series convergence
  • example series
    • geometric series
      • power series $\sum x^n$
    • harmonic series
    • $p$-series
  • divergence test
  • integral test
    • remainder estimate for integral test
  • comparison test
  • limit comparison test
  • alternating series
    • alternating series test
    • alternating series estimation theorem
  • absolute convergence
    • conditional convergence
    • ratio test
    • root test
  • convergence test strategy
  • power series
    • radius of convergence $R$
      • interval of convergence
      • find interval of convergence
    • differentiation or integration of power series
      • convergence and differentiability of power series
    • function $f(x)$’s power series expansion
  • Taylor series
    • Maclaurin series
    • $n$-th degree Taylor polynomial $T_n$
    • remainder of Taylor series $R_n$
      • Taylor’s inequality
      • Maclaurin series for $e^x$
      • Maclaurin series for trigonometric function
      • binomial series

Sequence and Series

sequence $\{a_n\}$

list of number in definite order

sequence convergence

$$\underset{n\to \infty }{\lim }{a}_n=L$$

$\Rightarrow \{a_n\}$ converge

• $\iff \forall \epsilon >0,\exists N,\forall n>N$

  $$|a_n-L|<\epsilon$$

• otherwise, $\{a_n\}$ diverge

prove sequence convergence by function

sequence converge if corresponding function converge

$$\begin{gathered} \underset{x\to \infty }{\lim }f(x)=Lf(n)=a_n(n\in \mathbb{N}) \\ \Rightarrow \underset{n\to \infty }{\lim }{a}_n=L \end{gathered}$$

• all limit law on function can be applied on sequence

sequence convergence preserve by chaining continuous function

$$\begin{gathered} \underset{n\to \infty }{\lim }{a}_n=Lf\text{ continuous at }L \\ \Rightarrow \underset{x\to \infty }{\lim }f(a_n)=f(L) \end{gathered}$$

convergence of $\{r^n\}$

$$\underset{n\to \infty }{\lim }{r}^n=\{\begin{array}{ll}0& r\in (-1,1)\\ 1& r=1\\ \text{DNE}& \text{otherwise}\end{array}$$

sequence monotonicity

increasing sequence

$$\forall n\ge 1,a_n<a_{n+1}$$

• the opposite is decreasing sequence

bounded sequence

$\{a_n\}$ bounded above

$$\iff \exists M,\forall n\ge 1,a_n<M$$

• the opposite is bounded below
• bounded sequence: both bounded above and below

monotonic sequence theorem

monotonic bounded sequence converge

series (infinite series)

$$\sum a_n$$

or

$$\sum _{n-1}^{\infty }{a}_n$$

$n$th partial sum $s_n$

$$s_n=\sum _{i=1}^n$$

series convergence

$$\underset{n\to \infty }{\lim }{s}_n=s$$

$\Rightarrow \sum a_n$ convergent

• $s$ sum of series
• the opposite is divergent
• linear combination of convergent series is convergent

example series

geometric series

$$\begin{gathered} a_n=a{r}^{n-1} \\ {s}_n=\frac{a(1-r^n)}{1-r} \\ \sum a_n=\{\begin{array}{ll}\frac{a}{1-r}& -1<r<1\\ \text{DNE}& \text{otherwise}\end{array} \end{gathered}$$

power series $\sum x^n$

$$\sum _{n=0}^{\infty }{x}^n=\frac{1}{1-x}$$

harmonic series

$$a_n=\frac{1}{n}$$

• divergent

  $$\sum _{n=1}^{\infty }\frac{1}{n}=\infty$$

$p$-series

$$\sum _{n=1}^{\infty }\frac{1}{n^p}\{\begin{array}{ll}\text{converge}& p>1\\ \text{diverge}& p\le 1\end{array}$$

• $p=2$

$$\sum _{n=1}^{\infty }\frac{1}{n^2}=\frac{{\pi }^2}{6}$$

divergence test

$\sum a_n$ converge $\Rightarrow \underset{n\to \infty }{\lim }{a}_n=0$

$\underset{n\to \infty }{\lim }{a}_n\ne 0\text{ or DNE}\Rightarrow \sum a_n$ diverge

integral test

$\exists N\ge 1\in {\mathbb{N}}^{+},f(n)=a_n$ continuous decreasing positive on $[N,\infty )$

${\int }_1^{\infty }f(x)dx$ converge $\iff \sum a_n$ converge

remainder estimate for integral test

$f$ decreasing on $[n,\infty )$ $$\begin{gathered} R_n=s-s_n \\ {\int }_{n+1}^{\infty }f(x)dx\le R_n\le {\int }_n^{\infty }f(x)dx \end{gathered}$$

comparison test

$\sum b_n$ converge, $\forall n,a_n\le b_n$
$\Rightarrow a_n$ converge

$\sum b_n$ diverge, $\forall n,b_n\le a_n$
$\Rightarrow a_n$ diverge

limit comparison test

$$\underset{n\to \infty }{\lim }\frac{a_n}{b_n}=\{\begin{array}{ll}C\ne 0& \Rightarrow a_n,b_n\text{ have same convergence}\\ 0,b_n\text{ converge}& \Rightarrow a_n\text{ converge}\\ \infty ,b_n\text{diverge}& \Rightarrow a_n\text{ diverge}\end{array}$$

alternating series

$$\forall n\ge 1\in \mathbb{N},a_n\cdot a_{n+1}<0$$

alternating series test

alternating series $a_n$

$$\begin{array}{l}\forall n\ge 1\in \mathbb{N},|a_n|\le |a_{n+1}|\\ \underset{a_n\to \infty }{\lim }=0\end{array}\}\Rightarrow a_n\text{ converge}$$

alternating series estimation theorem

$a_n$ satisfy alternating series test

$$|R_n|=|s-s_n|\le |a_{n+1}|$$

absolute convergence

$$\sum |a_n|\text{ converge}$$

$\Rightarrow \sum a_n$ absolutely converge

• $\Rightarrow \sum a_n$ converge

conditional convergence

$\sum a_n$ converge but not absolutely converge

$\Rightarrow \sum a_n$ conditionally converge

ratio test

$$\underset{n\to \infty }{\lim }\left|\frac{a_{n+1}}{a_n}\right|=L\{\begin{array}{l}<1\Rightarrow \sum a_n\text{ absolutely converge}\\ >1\text{ or }\infty \Rightarrow \sum a_n\text{ diverge}\end{array}$$

root test

$$\underset{n\to \infty }{\lim }\sqrt[n]{|a_n|}=L\{\begin{array}{l}<1\Rightarrow \sum a_n\text{ absolutely converge}\\ >1\text{ or }\infty \Rightarrow \sum a_n\text{ diverge}\end{array}$$

convergence test strategy

1. special series
   1. $p$-series
   2. geometric series
2. similar form to special series
   $\Rightarrow$ (limit) comparison test
3. divergence test
4. alternating series test
5. absolute convergence
   1. ratio test
   2. $(b_n{)}^n\Rightarrow$ root test
6. integral test

power series

power series centered at $a$ (power series about $a$)

$$\sum _{n=0}^{\infty }{c}_n(x-a{)}^n$$

• $c_n$ coefficient

radius of convergence $R$

$a_n$ converge when $x$ is within $R$ to $a$

1. $R=0$
   converge iff $x=a$
2. $R=\infty$
   converge for $x\in \mathbb{R}$
3. $R=C>0$
   converge if $|x-a|<R$, diverge if $|x-a|>R$

interval of convergence

1. $\{a\}$

2. $\mathbb{R}$

3. ⏎

   • $[a-R,a+R]$
   • $[a-R,a+R)$
   • $(a-R,a+R]$
   • $(a-R,a+R)$

find interval of convergence

1. find radius of convergence using ratio test/ root test
2. check endpoint

differentiation or integration of power series

equal to each term different or integrate

convergence and differentiability of power series

power series $$\sum a_n=\sum _{n=0}^{\infty }{c}_n(x-a{)}^n$$

has radius of convergence $R$

$\Rightarrow$ function

$$f(x)=\sum _{n=0}^{\infty }{c}_n(x-a{)}^n$$

differentiable on $(a-R,a+R)$

1. derivative

   $$\begin{gathered} f^{\prime }(x)=\sum _{n=0}^{\infty }\frac{d}{dx}\left(c_n(x-a{)}^n\right) \\ =\sum _{n=1}^{\infty }{c}_{n}n(x-a{)}^{n-1} \end{gathered}$$

   • $n$ start from $1$ because the term at $0$ is $0$

2. integral

   $$\begin{gathered} \int f(x)dx=\sum _{n=0}^{\infty }\int c_n(x-a{)}^{n}dx \\ =\sum _{n=0}^{\infty }{c}_n\frac{(x-a{)}^{n+1}}{x+1}+C \end{gathered}$$

function $f(x)$’s power series expansion

$$f(x)=\sum _{n=0}^{\infty }{c}_n(x-a{)}^n|x-a|<R$$

$\iff$

$$c_n=\frac{f^{(n)}(a)}{n!}$$

Taylor series

Taylor series of $f$ at $a$ (/ about $a$ / centered at $a$)

$$f(x)=\sum _{n=0}^{\infty }\frac{f^{(n)}(a)}{n!}(x-a{)}^n$$

Maclaurin series

Taylor series at $0$

$$f(x)=\sum _{n=0}^{\infty }\frac{f^{(n)}(0)}{n!}{x}^n$$

$n$-th degree Taylor polynomial $T_n$

$n$-th degree Taylor polynomial $T_n$ of $f$ at $a$

$$T_n(x)=\sum _{i=0}^n\frac{f^{(n)}(a)}{n!}(x-a{)}^n$$

remainder of Taylor series $R_n$

$$R_n(x)=f(x)-T_n(x)$$

• for $|x-a|<R$

  $$\underset{n\to \infty }{\lim }{R}_n(x)=0$$

  $\Rightarrow f$ equal sum of its Taylor series

Taylor’s inequality

$|f^{(n+1)}(x)|\le M$ for $|x-a|\le d$

$$\Rightarrow |R_n(x)|\le \frac{M}{(n+1)!}|x-a{|}^{n+1}$$

for $|x-a|\le d$

• used to prove equivalence between function and sum of Taylor series

• a convergent sequence $(x\in \mathbb{R})$

  $$\underset{n\to \infty }{\lim }\frac{x^n}{n!}=0$$

Maclaurin series for $e^x$

$$e^x=\sum _{n=0}^{\infty }\frac{x^n}{n!}$$

• let $x=1$ $$e=\sum _{n=0}^{\infty }\frac{1}{n!}$$

Maclaurin series for trigonometric function

$$\begin{gathered} \sin x=\sum _{n=0}^{\infty }(-1{)}^n\frac{x^{2n+1}}{(2n+1)!} \\ \cos x=\sum _{n=0}^{\infty }(-1{)}^n\frac{x^{2n}}{(2n)!} \end{gathered}$$

binomial series

$|x|<1$

$$(1+x{)}^k=\sum _{n=0}^{\infty }(\begin{array}{c}k\\ n\end{array})x^n$$