# 物理代写|理论力学作业代写Theoretical Mechanics代考|Sequence of Numbers and Limiting Values

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Sequence of Numbers and Limiting Values

By a sequence of numbers we will understand a sequence of (indexed) real numbers:
$$a_1, a_2, a_3, \cdots, a_n, \cdots \quad a_n \in \mathbb{R}$$
We have finite and infinite sequences of numbers. In case of a finite sequence the index $n$ is restricted to a finite subset of $\mathbb{N}$. The sequence is formally denoted by the symbol
$$\left{a_n\right}$$
and represents a mapping of the natural numbers $\mathbb{N}$ on the body of real numbers $\mathbb{R}$ :
$$f: n \in \mathbb{N} \longrightarrow a_n \in \mathbb{R} \quad\left(n \longrightarrow a_n\right)$$
Examples
1.
$$a_n=\frac{1}{n} \quad \longrightarrow a_1=1, a_2=\frac{1}{2}, a_3=\frac{1}{3}, a_4=\frac{1}{4} \cdots$$

2.
$$a_n=\frac{1}{n(n+1)} \quad \longrightarrow a_1=\frac{1}{1 \cdot 2}, a_2=\frac{1}{2 \cdot 3}, a_3=\frac{1}{3 \cdot 4}, \cdots$$
3.
$$a_n=1+\frac{1}{n} \quad \longrightarrow a_1=2, a_2=\frac{3}{2}, a_3=\frac{4}{3}, a_4=\frac{5}{4}, \cdots$$
Now we define the
Limiting value (limit) of a sequence of numbers
If $a_n$ approaches for $n \rightarrow \infty$ a single finite number $a$, then $a$ is the limiting value (limes) of the sequence $\left{a_n\right}$ :
$$\lim _{n \rightarrow \infty} a_n=a ; a_n \stackrel{n \rightarrow \infty}{\longrightarrow} a .$$

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Series and Limiting Values

Adding up the terms of an infinite sequence of numbers leads to what is called a series:
$$a_1, a_2, a_3, \cdots, a_n, \cdots \curvearrowright a_1+a_2+a_3+\cdots+a_n+\cdots=\sum_{m=1}^{\infty} a_m$$

Strictly, the series is defined as limiting value of a sequence of (finite) partial sums:
$$S_r=\sum_{m=1}^r a_m .$$
The series converges to $S$ if
$$\lim {r \rightarrow \infty} S_r=S$$ does exist. If not then it is called divergent. A necessary condition for the series $\sum{m=1}^{\infty} a_m$ to be convergent is
$$\lim {m \rightarrow \infty} a_m=0$$ For, if $\sum{m=1}^{\infty} a_m$ is indeed convergent then it must hold:
$$\lim {m \rightarrow \infty} a_m=\lim {m \rightarrow \infty}\left(S_m-S_{m-1}\right)=\lim {m \rightarrow \infty} S_m-\lim {m \rightarrow \infty} S_{m-1}=S-S=0$$

# 理论力学代写

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Sequence of Numbers and Limiting Values

$$a_1, a_2, a_3, \cdots, a_n, \cdots \quad a_n \in \mathbb{R}$$

$$f: n \in \mathbb{N} \longrightarrow a_n \in \mathbb{R} \quad\left(n \longrightarrow a_n\right)$$

1.
$$a_n=\frac{1}{n} \quad \longrightarrow a_1=1, a_2=\frac{1}{2}, a_3=\frac{1}{3}, a_4=\frac{1}{4} \cdots$$
2.
$$a_n=\frac{1}{n(n+1)} \quad \longrightarrow a_1=\frac{1}{1 \cdot 2}, a_2=\frac{1}{2 \cdot 3}, a_3=\frac{1}{3 \cdot 4}, \cdots$$
3.
$$a_n=1+\frac{1}{n} \quad \longrightarrow a_1=2, a_2=\frac{3}{2}, a_3=\frac{4}{3}, a_4=\frac{5}{4}, \cdots$$

$$\lim _{n \rightarrow \infty} a_n=a ; a_n \stackrel{n \rightarrow \infty}{\longrightarrow} a .$$

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Series and Limiting Values

$$a_1, a_2, a_3, \cdots, a_n, \cdots \curvearrowright a_1+a_2+a_3+\cdots+a_n+\cdots=\sum_{m=1}^{\infty} a_m$$

$$S_r=\sum_{m=1}^r a_m .$$

$$\lim r \rightarrow \infty S_r=S$$

$$\lim m \rightarrow \infty a_m=0$$

$$\lim m \rightarrow \infty a_m=\lim m \rightarrow \infty\left(S_m-S_{m-1}\right)=\lim m \rightarrow \infty S_m-\lim m \rightarrow \infty S_{m-1}$$

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