Calculus: Series

# WHY: Why should we learn Series

Making Linear to Discrete.
A tool for solving some problems in Physics(TAYLOR POLYNOMIALS, etc.) and some situation.

# WHAT: What is Sequence

Sequences: A list of number written in a definite order $\{a_n\}_{n = 1}^\infty$

# Convergence

$\forall \epsilon, \exists N, \forall n > N, |a_n - l| < \epsilon \Rightarrow a_n \rightarrow l$ OR $\lim_{n \rightarrow \infty}a_n = L \Rightarrow a_n \rightarrow L$

E.g. $1/x$: for all $\epsilon$, exist $N = \frac{1}{\epsilon}$, for all $n > N, |a_n - 0| < \epsilon \Rightarrow a_n$ convergent to 0

Test
- Bounded by lines: Bounded in both sides + Monotonically increasing or decreasing (the continuous decreasing of distance) = Convergent
  - Bounded in both sides: $\forall n, a_n \leq M$ and $\forall n, a_n \geq N$
  - Monotonically increasing or decreasing: $\forall n, a_n < a_{n+1}$ or $\forall n, a_n > a_{n+1}$
    
    Note. Identifying between decreasing and non-increasing
- Bounded by functions (Squeeze Theorem): $a_n \leq b_n \leq c_n \Rightarrow \lim{b_n} = L$
Usage
- Continuity and Convergence theorem: $\lim_{x \rightarrow L}f \exists, \lim_{n \rightarrow \infty}a_n = L \Rightarrow \lim_{n \rightarrow \infty}f$
  - $f(x): a_n = f(n), \lim_{x \rightarrow \infty}f = L \Rightarrow a_n \rightarrow L$
  - $a_n, b_n, d_n \rightarrow C \Rightarrow \lim{(a^C + b \cdot d)} = (\lim{a})^C + \lim{b} \cdot \lim{d}$

Exercise.

$$ \text{Find if (a) $a_n = \frac{n!}{n^n} (a > 0)$, (b) $a_n = \frac{a^n}{n!} (a > 0)$, (c) $a_n = \frac{n^b}{a^n}$, (d) $a_n = \frac{\ln n}{n^b}$ converges} $$

$n^n \gg n! \gg a^n \gg n^b \gg \ln{n} \gg C$

$$ \text{(a) $a_n = (1 + \frac{1}{n})^n$, (b) $a_n = (\frac{n}{n+1})^{2n}$ converges} $$ $$ \text{Hint 1: } (1 - x)^n \geq 1 - nx, (1 + x)^n \geq 1 + nx, x \in [0, 1] $$ $$ \text{Hint 2: } a_n < b_n = (1 + \frac{1}{n})^{n + 1} $$

# Arbitrary Sequence

Arithmetic sequence
Geometric progression

# WHAT: What is Series

Series: the sum of the infinite terms from a sequence (limit the $n$ of partial sum $S_n = \sum_{i = 1}^na_n$ to infinity) “The Integral of a Sequence”

# Convergence: The divergence or convergence of $S_n \Leftrightarrow Series$ diverges or converges

Test for Series
- Direct Test
  - Overall elements (Integral Test): $f(x)$ is continuous and $a_n = f(n)$, $\sum a_n \rightarrow/\nrightarrow C \Leftrightarrow \int_1^\infty f(x)\,\mathrm{d}x \rightarrow/\nrightarrow C$
  - Adjacent elements (Ratio test): $\lim_{n \rightarrow \infty}|\frac{a_{n + 1}}{a_n}| = l < 1 \Rightarrow$ absolute converge (prove using geometric series) (> 1 - diverge, = 1 - inconclusive)
  - Single elements (Root Test): $\lim_{n \rightarrow \infty}\sqrt[n]{|a_n|}$
- Non-direct Test: Compare with another
  - Limit Comparison test: $\sum a_n, \sum b_n > 0, \lim_{n \rightarrow \infty}\frac{a_n}{b_n} = C > 0 \Rightarrow \sum a_n, \sum b_n$ both di/converge.
  - Comparison test(“Squeeze Theorem”): $\sum a_n, \sum b_n > 0, \sum b_n$ converges and $a_n \leq b_n$ $\Rightarrow$ $\sum a_n$ converges, vice versa (con to di, $\leq$ to $\geq$).
    - Absolutely converge $\sum|a_n|$ is converges (Conditionally = the condition is not plug in absolute value): $0 \leq \sum a_n + \sum|a_n| \leq 2 \sum|a_n|$
Usage
- $\sum a_n, \sum b_n$ converge, $\sum a_n + \sum b_n = \sum (a_n + b_n)$ and $\sum Ca_n = C \sum a_n$
The relationship of Sequence and Series
- series $\sum_{n = 1}^\infty$ converges $\Rightarrow$ $\lim_{n \rightarrow \infty}a_n = 0$, or series $\sum_{n = 1}^\infty$ diverges $\Rightarrow$ $\lim_{n \rightarrow \infty}a_n \neq 0$
  - minus prove: $S_{n} - S_{n - 1}$
  - integral prove
- $\lim_{n \rightarrow \infty}a_n = 0$ $\nRightarrow$ series $\sum_{n = 1}^\infty$ converges, e.g. $\sum_{i = 1}^\infty\frac{1}{i} > \ln{(n + 1)}$

Exercise.

$$ \sum{\frac{1}{n^{1+\frac{1}{n}}}}, \sum \frac{1}{n^{1 + f(n)}}, \sum{\sin{\frac{1}{n}}} $$ $$ \sum(1 - \frac{1}{n})^n, \sum(1 - \frac{1}{n})^{n^2}, \sum{\ln(1 + \frac{1}{n})}, \sum \ln(1 + \frac{1}{f(x)}) $$ $$ \sum \frac{x^n}{\ln^p x}, \text{Known $b_n \rightarrow \frac{2}{5}$, }\sum_1\frac{(-1)^nn!}{n^nb_1b_2\ldots b_n} $$

# Arbitrary Series

Geometric Series
Alternating Series $\sum(-1)^{n - 1} a_n$
- $a_{n + 1} \leq a_n, \lim_{n \rightarrow \infty}a_n = 0 \Rightarrow \sum(-1)^{n - 1} a_n \rightarrow C$ (decrease + bounded): $S_{2n} = (a_1 - a_2) + (a_3 - a_4) + \ldots +(a_{2n - 1} - a_{2n}) > 0; S_{2n} = a_1 - (a_2 - a_3) - \ldots - a_{2n} < a_1$, then same as $S_{2n + 1} \Rightarrow$ whether n is odd or even, series converges.
- Alternating Series Estimation Theorem: As the same way as above, we also could derive that $S_{2n} = (a_1 - a_2) + (a_3 - a_4) + \ldots +(a_{2n - 1} - a_{2n}) > (a_1 - a_2) + (a_3 - a_4) + \ldots; S_{2n} = S_{2n} = a_1 - (a_2 - a_3) - \ldots - a_{2n} < a_1 - (a_2 - a_3) - \ldots$

# HOW

# WHAT: What is the applycation of series - Power Series

$f(x) = \sum c(x - a)^n$ is called a power series centered at $a$, domain of $f(x)$ is the set that series converge.

only converge at $x = a$
converge for all $x \in R$
$\exists R > 0, |x - a| < R \text{ the series converge}, R: \text{radius of convergence}$

# HOW: Ratio Test + Two sides

Exercise.

$$ \text{Find the domain of } f(x) = \sum \frac{(x - 3)^n}{n} (ANS: [2,4)) $$ $$ \text{} $$

# HOW: How to apply

if the power series has a convergent radius R > 0, the f(x) is both differentiatable and integrable on the interval (a - R, a + R), and

$$ f’(x) = \sum n c_n (x - a)^{n - 1} $$ $$ \int f(x)\,\mathrm{d}x = c + \sum \frac{c_n}{n + 1}(x - a)^{n + 1} $$

E.g.

$$ |x| < 1, \frac{1}{1 - x} = \sum x^n \Rightarrow (\frac{1}{1 - x})^2 = \sum nx^{n - 1} \Rightarrow -\ln{(1 - x)} = \sum \frac{x^{n + 1}}{n + 1} $$ $$ f(x) = \frac{1}{1 - x} \Rightarrow f^{(10)}(x)|\_{x = 0} = 10! $$

Exercise.

$$ \text{Expand $\int{\frac{x}{8+x^3}}\,\mathrm{d}x$ into power series} $$ $$ \text{If $f(x) = \sum c_n x^n$, where $c_{n + 4} = c_n$ for all $n \geq 0$,} $$ $$ \text{find the interval of convergence of the series and the formula for f(x)} $$

# HOW: How to apply for graphing - Taylor Series

$$ c_n = \frac{f^{(n)}(a)}{n!} \Rightarrow f(x) = \sum \frac{f^{(n)}(a)}{n!}(x - a)^n, |x - a| < R $$

when $a = 0$, it is called Maclaurin series

$1 + x$ Format $$ \frac{1}{1 - x}, |x| < 1 = 1 + x + x^2 + x^3 \ldots $$ $$ \ln(1 + x) = \int \frac{1}{1 + x}\,\mathrm{d}x = \sum \frac{(-1)^nx^{n + 1}}{n + 1} = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} \ldots $$ $$ (1 + x)^k = \sum \begin{pmatrix}k \\ n\end{pmatrix}x^n \begin{cases} |x| < 1 & k \in \mathbb{R}: \text{abc} \\ |x| = 1 & k \geq 0: \text{abc}, k \geq -1: \text{con} \\ |x| = -1 & k \geq 0: \text{con} \\ |x| > 1 & \text{di unless $k$ is non negint} \end{cases} $$
Trigonometrical Format $$ \sin{x} = \sum(-1)^{n}\frac{x^{2n + 1}}{(2n + 1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots $$ $$ \cos{x} = \frac{\mathrm{d}}{\mathrm{d}x}\sin{x} = \sum (-1)^n \frac{x^{2n}}{2n} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \ldots $$
Special Format $$ e^x = \sum \frac{1}{n!}x^n = \sum\frac{e^a}{n!}(x - a)^n = 1 + x + \frac{1}{2!}x^2 + \frac{1}{3!}x^3 + \ldots $$ $$ \arctan{x} = \int \frac{1}{1 + x^2} \,\mathrm{d}x = \sum \frac{(-1)^nx^{2n + 1}}{2n +1} = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \ldots $$

Notice. $\int$ and $\sum$ cannot always exchange the places for some specific infinite sum.

Exercise.

$$ \text{Known $\int_0^\infty e^{-x} x^n \,\mathrm{d}x = n!$, derive $\frac{1}{\pi}\int_0^1\frac{\sin{\ln{x}}}{\ln{x}}\,\mathrm{d}x$} $$ $$ \text{Find if and how the series is convergent: } \sum(-1)^n \frac{\tan^{-1}n}{n} $$ $$ \text{Expend $f(t) = \frac{1 - tx}{1 - 2tx - t^2}$ to power series} $$ $$ \text{Find the compact form for of $\sum \frac{(\frac{1}{2})^n x^{2n}}{(n + 1)!}$} $$

What about the Error Bond if the series finite?

$$ f(x) = \sum_0^N\frac{f^{n}(a)}{n!}(x - a)^n + \sum_{N + 1}^\infty\frac{f^{n}(a)}{n!}(x - a)^n = T_N(x) + R_N(x) $$ $$ \text{Accordint to MVT, } \exists z \in [x, a], \frac{f^{N + 1}(z)}{(N + 1)!}(x - a)^{N + 1} = R_N(x) $$ $$ \Rightarrow \exists z \in [x, a], [\frac{f^{N + 1}(z)}{(N + 1)!}(x - a)^{N + 1}]_{max} \geq R_N(x) $$