### 数学代写|微积分代写Calculus代写|Sequence and Series of Real Numbers

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## 数学代写|微积分代写Calculus代写|Sequence of Real Numbers

Suppose that, for each positive integer $n$, we are given a real number $a_{n}$. Then, the list of numbers
$$a_{1}, a_{2}, \ldots, a_{n}, \ldots$$
is called a sequence. A more precise definition of a sequence is the following:
Definition 1.1.1 A sequence of real numbers is a function from the set No natural numbers to the set $\mathbb{R}$ of real numbers.

Notation 1.1.1 If $f: \mathbb{N} \rightarrow \mathbb{R}$ is a sequence, and if $a_{n}=f(n)$ for $n \in \mathbb{N}$, then we write the sequence $f$ as
$$\left(a_{1}, a_{2}, \ldots\right) \text { or }\left(a_{n}\right) \text { or }\left{a_{n}\right},$$ and the term $a_{n}$ is called the $n^{\text {th }}$ term of the sequence $\left(a_{n}\right)$. In specific cases, where one knows an expression for $a_{n}$, one may write
$\left(a_{1}, a_{2}, \ldots, a_{n}, \ldots\right)$ instead of $\left(a_{1}, a_{2}, \ldots\right)$
Remark 1.1.1 It is to be borne in mind that a sequence $\left(a_{1}, a_{2}, \ldots\right)$ is different from the set $\left{a_{n}: n \in \mathbb{N}\right}$. For instance, a number may be repeated in a sequence $\left(a_{n}\right)$, but it need not be written repeatedly in the set $\left{a_{n}: n \in \mathbb{N}\right}$. As an example, $\left(1, \frac{1}{2}, 1, \frac{1}{3}, \ldots, 1, \frac{1}{n}, \ldots\right)$ is a sequence $\left(a_{n}\right)$ with $a_{2 n-1}=1$ and $a_{2 n}=1 /(n+1)$ for each $n \in \mathbb{N}$, whereas the set $\left{a_{n}: n \in \mathbb{N}\right}$ is same as the set ${1 / n: n \in \mathbb{N}}$.

Remark 1.1.2 We can also talk about a sequence of elements from any non-empty set $S$, such as sequence of sets, sequence of functions and so on. Thus, given a non-empty set $S$, a sequence in $S$ is a function $f: \mathbb{N} \rightarrow S$. In this chapter, we shall consider only sequence of real numbers. In some of the later chapters we shall consider sequences of functions as well.
Example 1.1.1 Let us consider a few examples of sequences (Fig. 1.1):
(i) $\left(a_{n}\right)$ with $a_{n}=1$ for all $n \in \mathbb{N}$.
(ii) $\left(a_{n}\right)$ with $a_{n}=n$ for all $n \in \mathbb{N}$.
(iii) (an) with $a_{n}=1 / n$ for all $n \in \mathbb{N}$.
(iv) $\left(a_{n}\right)$ with $a_{n}=n /(n+1)$ for all $n \in \mathbb{N}$.
(v) $\left(a_{n}\right)$ with $a_{n}=(-1)^{n}$ for all $n \in \mathbb{N}$. This sequence takes values 1 and $-1$ alternately.

## 数学代写|微积分代写Calculus代写|Convergence and Divergence

In certain sequences the $n^{\text {th }}$ term comes closer and closer to a particular number as $n$ becomes larger and larger. For example, in the sequence $\left(\frac{1}{n}\right)$, the $n^{\text {th }}$ term comes closer and closer to 0 , whereas in $\left(\frac{n}{n+1}\right)$, the $n^{\text {th }}$ term comes closer and closer to 1 as $n$ becomes larger and larger. If you look at the sequence $\left((-1)^{n}\right)$, the terms oscillate between $-1$ and 1 as $n$ varies, whereas in $\left(n^{2}\right)$ the terms become larger and larger.
Now, we make precise the the statement ” $a_{n}$ comes closer and closer to a number $a$ as $n$ becomes larger and larger”, that is, ” $a_{n}$ can be made arbitrarily close to $a$ by taking $n$ large enough”, by defining the notion of convergence of a sequence.

Definition 1.1.2 A sequence $\left(a_{n}\right)$ of real numbers is said to converge to a real number $a$ if for every $\varepsilon>0$, there exists a positive integer $N$, that may depend on $\varepsilon$, such that
$$\left|a_{n}-a\right|<\varepsilon \quad \forall n \geq N$$
A sequence that converges is called a convergent sequence, and a sequence that does not converge is called a divergent sequence.
Notation 1.1.2 (i) If $\left(a_{n}\right)$ converges to $a$, then we write
$$a_{n} \rightarrow a \text { as } n \rightarrow \infty$$
that we may read as ” $a_{n}$ tends to $a$ as $n$ tends to infinity”, that we also write in short as $a_{n} \rightarrow a$.
(ii) If $\left(a_{n}\right)$ does not converge to $a$, then we write $a_{n} \nrightarrow a$.
Remark 1.1.3 We must keep in mind that the symbol o is not a number; it is only a notation used in the context of describing some properties of real numbers, such as in Definition 1.1.2.

Remark 1.1.4 In Definition 1.1.2, the expression $\left|a_{n}-a\right|<\varepsilon$ can be replaced by $\left|a_{n}-a\right| \leq \varepsilon$ or by $\left|a_{n}-a\right|0$. In other words, the following statements are equivalent.
(i) For every $\varepsilon>0$, there exists $N \in \mathbb{N}$ such that $\left|a_{n}-a\right|<\varepsilon$ for all $n \geq N$. (ii) For every $\varepsilon>0$, there exists $N \in \mathbb{N}$ such that $\left|a_{n}-a\right| \leq \varepsilon$ for all $n \geq N$.
(iii) For every $\varepsilon>0$, there exists $N \in \mathbb{N}$ such that $\left|a_{n}-a\right| \leq c_{0} \varepsilon$ for all $n \geq N$ for some $c_{0}>0$.

Clearly, (i) implies (ii). To see (ii) implies (i), assume (ii) and let $\varepsilon>0$ be given. Then, by (ii), with $\varepsilon / 2$ in place of $\varepsilon$, there exists $N \in \mathbb{N}$ such that $\left|a_{n}-a\right| \leq \varepsilon / 2$ for all $n \geq N$. In particular, (i) holds. Now, (iii) follows from (i) by taking $c_{0} \varepsilon$ in place of $\varepsilon$, and (i) follows from (iii) by taking $\varepsilon / c_{0}$ in place of $\varepsilon$.

## 数学代写|微积分代写Calculus代写|Some Tests for Convergence and Divergence

Theorem 1.1.5 (Ratio test) Suppose $a_{n}>0$ for all $n \in \mathbb{N}$ such that $\lim {n \rightarrow \infty} \frac{a{n+1}}{a_{n}}=\ell$ for some $\ell \geq 0$. Then the following hold.
(i) If $\ell<1$, then $a_{n} \rightarrow 0$. (ii) If $\ell>1$, then $a_{n} \rightarrow \infty$.
Proof (i) Suppose $\ell<1$. Let $q$ be such that $\ell1$. Let $q$ be such that $1<q<\ell$. Then, taking for example the open interval $I$ containing $\ell$ as $I=(q, \ell+1)$, there exists $N \in \mathbb{N}$ such that $$\frac{a_{n+1}}{a_{n}}>q \quad \forall n \geq N$$
Hence,
$$a_{n} \geq q^{n-N} a_{N} \quad \forall n \geq N$$
By Theorem 1.1.4, $q^{n-N} \rightarrow \infty$ as $n \rightarrow \infty$. Hence, $a_{n} \rightarrow \infty$
Example 1.1.15 Let $0<a<1$. Then $n a^{n} \rightarrow 0$ as $n \rightarrow \infty$. To see this, let $a_{n}:=$ $n a^{n}$ for $n \in \mathbb{N}$. Then we have
$$\frac{a_{n+1}}{a_{n}}=\frac{(n+1) a^{n+1}}{n a^{n}}=\frac{(n+1) a}{n} \quad \forall n \in \mathbb{N} .$$
Hence, $\lim {n \rightarrow \infty} \frac{a{n+1}}{a_{n}}=a<1$. Thus, by Theorem 1.1.5, $n a^{n} \rightarrow 0$.
Similarly, it can be shown that, for any $k \in \mathbb{N}, n^{k} a^{n} \rightarrow 0$ as $n \rightarrow \infty$.
Remark 1.1.9 The converse of the results (i) and (ii) in Theorem 1.1.5 does not hold. To see this, consider the following examples:
(i) Consider $\left(a_{n}\right)$ with $a_{n}=1 / n$. Then $a_{n} \rightarrow 0$, but $a_{n+1} / a_{n} \rightarrow 1$.
(ii) Consider $\left(a_{n}\right)$ with $a_{n}=n$. Then $a_{n} \rightarrow \infty$, but $a_{n+1} / a_{n} \rightarrow 1$.
In fact, we shall see in the next section that the condition $\ell<1$ in Theorem 1.1.5 is too strong, in the sense that, not only we have the convergence of $\left(a_{n}\right)$ to 0 , but also we can show the convergence of the sequence $\left(s_{n}\right)$, where $s_{n}=a_{1}+a_{2}+\cdots+a_{n}$.
Note that, for any sequence $\left(a_{n}\right)$, if $s_{n}=a_{1}+a_{2}+\cdots+a_{n}$, then the convergence of $\left(s_{n}\right)$ implies $a_{n} \rightarrow 0$, but $a_{n} \rightarrow 0$ does not imply the convergence of $\left(s_{n}\right)$.

Exercise 1.1.10 Establish the statement in the last paragraph of the above remark. $\varangle$
The following theorem gives a sufficient condition for certain number to be a limit of a given sequence.

## 数学代写|微积分代写Calculus代写|Sequence of Real Numbers

\left(a_{1}, a_{2}, \ldots\right) \text { 或 }\left(a_{n}\right) \text { 或 }\left{a_{n}\right},\left(a_{1}, a_{2}, \ldots\right) \text { 或 }\left(a_{n}\right) \text { 或 }\left{a_{n}\right},和术语一个n被称为nth 序列项(一个n). 在特定情况下，人们知道一个表达式一个n, 可以写
(一个1,一个2,…,一个n,…)代替(一个1,一个2,…)

(i)(一个n)和一个n=1对所有人n∈ñ.
(二)(一个n)和一个n=n对所有人n∈ñ.
(iii) (an) 与一个n=1/n对所有人n∈ñ.
(四)(一个n)和一个n=n/(n+1)对所有人n∈ñ.
（在）(一个n)和一个n=(−1)n对所有人n∈ñ. 这个序列取值 1 和−1交替。

## 数学代写|微积分代写Calculus代写|Convergence and Divergence

|一个n−一个|<e∀n≥ñ

(ii) 如果(一个n)不收敛到一个，然后我们写一个n↛一个.

(一) 对于每个e>0， 那里存在ñ∈ñ这样|一个n−一个|<e对所有人n≥ñ. (ii) 对于每个e>0， 那里存在ñ∈ñ这样|一个n−一个|≤e对所有人n≥ñ.
(iii) 对于每个e>0， 那里存在ñ∈ñ这样|一个n−一个|≤C0e对所有人n≥ñ对于一些C0>0.

## 数学代写|微积分代写Calculus代写|Some Tests for Convergence and Divergence

(一) 如果ℓ<1， 然后一个n→0. (ii) 如果ℓ>1， 然后一个n→∞.

(i) 考虑(一个n)和一个n=1/n. 然后一个n→0， 但一个n+1/一个n→1.
(ii) 考虑(一个n)和一个n=n. 然后一个n→∞， 但一个n+1/一个n→1.

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