Chapter 8 Infinite Sequences and Series

# Sequences

• A sequence is a function whose domain is $N$. ($N$ is the set of all positive integers.)
• In notation, we write $f(n) = a_n$.

# Theorem

1. $lim_{x \to \infty}f(x) = L, x \in R \Rightarrow lim_{n \to \infty}f(n) = L,n \in N$.
2. $lim_{n \to \infty}| a_n | = 0 \Rightarrow lim_{n \to \infty} a_n = 0$.
3. $\{ r^n \}^\infty_{n=1}$ converges iff $-1 < r \le 1$.
4. Let $\{ a_n \}$ be monotonic (單調) and bounded (有界) $\Rightarrow \{ a_n \}$ converges.

# Series

• Series: $\sum^{\infty}_{i=1} a_i = a_1 + a_2 + ... +a_n + ...$
• The convergence/divergence of a series：(n-th) partial sum = $S_n = \sum^{n}_{i=1} a_i$，
  $\{ S_n \}$ converges/diverges $\Leftrightarrow \sum^{\infty}_{i=1} a_i$ converges/diverges.

# Theorem (Geometric Series.)

$\sum^{\infty}_{i=1}r^i$ converges $\Leftrightarrow |r| < 1$ and $\sum^{\infty}_{i=1}r^i = \frac{r}{1-r}$.

# Theorem

$\sum^{\infty}_{i=1} a_i$ converges $\Rightarrow lim_{n \to \infty} a_n = 0$

# The Integral Test and Estimates of Sums

What kind of series can apply the integral test to check its convergence?
$a_i$ satisfy:
(i) essentially positive
(ii) decreasing
(iii) $f(x) = a_x$ is continuous

# The Integral Test

• $\sum^{\infty}_{n=1} \frac{1}{n^p}$ convergences, $p>1$
• $\sum^{\infty}_{n=1} \frac{1}{n^p}$ divergences, $p \le 1$

# Estimates of Sums

Let $\{a_i\}$ can apply the integral test, and let $R_n = S - S_n$,
Here $S = \sum^{\infty}_{i=1} a_i$ and $S_n = \sum^{n}_{i=1} a_i$
Then $\int^{\infty}_{n+1} a_x dx \le R_n \le \int^{\infty}_n a_x dx$

# The Comparison Tests

• 減法比較:
  • $a_n \ge b_n$ and $\sum a_n$ converges $\Rightarrow \sum b_n$ converges.
  • $a_n \ge b_n$ and $\sum b_n$ diverges $\Rightarrow \sum b_n$ diverges.
• 除法比較:
  • $lim_{n \to \infty} \frac{a_n}{b_n} = c > 0$
    $\Rightarrow$ Both $\sum a_n$ and $\sum b_n$ converge and diverge.
  • $lim_{n \to \infty} \frac{a_n}{b_n} = 0$
    • If $\sum b_n$ converges $\Rightarrow \sum a_n$ converges.
    • If $\sum a_n$ diverges $\Rightarrow \sum b_n$ diverges.

# Alternating Series

# Theorem

1. $a_n$ is decreasing.
2. $a_n \to 0$ as $n \to \infty$.
   $\Rightarrow \sum^{\infty}_{n=1}(-1)^{n+1}a_n$ converges.

# Error Estimate

$\sum^{\infty}_{n=1}(-1)^{n+1}a_n$ satisfies 1. and 2.
$|R_n| = |S - S_n| \le |S_{n+1} - S_n| \le a_{n+1}$.

# Absolute Convergence and The Ratio and Root test

# Define

1. $\sum_{n=1}^{\infty} a_n$ is said to be absolute convergence (AC) if $\sum_{n=1}^{\infty} |a_n|$ is convergent.
2. If $\sum_{n=1}^{\infty} a_n$ is convergent but not AC,then $\sum_{n=1}^{\infty} a_n$ is called conditional convergence (CC).

# Theorem

AC $\Rightarrow$ convergence.

# Ratio Test

比前後像縮小的比例

• $lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = L$
• 適用對象：有 "!" 項。

# Root Test

開$n$ 次根號也為檢查尾巴項的行為

• $lim_{n \to \infty} \sqrt[n]{|a_n|} = L$
• 適用對象：有\sqrt[n] 或$()^n$ 項。

# Conclusion

1. $L < 1 \Rightarrow$ AC.
2. $L > 1 \Rightarrow$ Divergence.
3. $L = 1 \Rightarrow$ The test fails.

# Power Series

# Define

$\sum_{n=0}^{\infty}c_n(x-a)^n = c_0+c_1(x-a)+c_2(x-a)^2 + ...$ is called a power series about $a$(or in $(x-a)$).

# Theorem

For series $\sum^{\infty}_{n=1}c_n(x-a)^n$,either one of the following 3 possibilities holds:

1. $r = 0$: 只有一點收斂
2. $r < \infty$: 兩端點須被檢驗，才可知其收斂與否
3. $r = \infty$: 每一點皆收斂

# Representations of Functions as Power Series

# Using Geometric Series

$\frac{1}{1-x} = 1+x+x^2+... = \sum_{n=0}^{\infty}x^n, |x| <1$.

# Taylor and Maclaurin Series

# Define

1. Taylor series of $f$ at $x = a$ is defined to be $\sum_{k=0}^{\infty}\frac{f^(k)(a)}{k!}(x-a)^k$.
2. If $a=0$,the corresponding Taylor series is called Maclaurin series.
3. $\sum_{k=0}^{\infty}\frac{f^(k)(a)}{k!}(x-a)^k$ is called the $n$ th-degree Taylor polynomial of $f$ at $a$.

# Taylor Inequality

Let $f(x)-T_n(x) = R_n(x)$,where $T_n = \sum_{k=0}^{n}\frac{f^(k)(a)}{k!}(x-a)^k$ is called the n-th degree Taylor polynomial of $f$ at $a$.

If $|f^{(n+1)}(x)| \le M$ for $|x-a| \le d$, then $|R_n(x)| \le \frac{M}{(n+1)!}|x-a|^{n+1}$ for $|x-a| \le d$.

# Reference

• 莊重 - 微積分 (二) Calculus II - 103 學年度