## 数学代写|实分析作业代写Real analysis代考|Cauchy’s Form for the Remainder

Under the additional assumption of continuity of $f^{(n+1)}$ we obtain Cauchy’s form for the remainder as follows:

COROLLARY 8.7.19 Let $f$ be a real-valued function on an open interval $I, c \in I$ and $n \in \mathbb{N}$. If $f^{(n+1)}$ is continuous on $I$, then for each $x \in I$, there exists a $\zeta$ between $c$ and $x$ such that
$$R_n(x)=R_n(f, c)(x)=\frac{f^{(n+1)}(\zeta)}{n !}(x-c)(x-\zeta)^n$$
Proof. Since $f^{(n+1)}(t)(x-t)^n$ is continuous on the interval from $c$ to $x$, by the mean value theorem for integrals (Theorem 6.3.6), there exists a $\zeta$ between $c$ and $x$ such that
$$\int_c^x f^{(n+1)}(t)(x-t)^n d t=(x-c) f^{(n+1)}(\zeta)(x-\zeta)^n$$
The result now follows by (20).
We now compute the Taylor series for several elementary functions and use the previous formulas for the remainder to show that the series converges to the function.

EXAMPLES 8.7.20 (a) As our first example we prove the binomial theorem (Theorem 3.2.5). For $n \in \mathbb{N}$ let $f(x)=(1+x)^n, x \in \mathbb{R}$. Since $f$ is a polynomial of degree $n$, if $k>n$ then $f^{(k)}(x)=0$ for all $x \in \mathbb{R}$. Therefore by Theorem 8.7.16,
$$f(x)=\sum_{k=0}^n \frac{f^{(k)}(0)}{k !} x^k$$
By computation, $f^{(k)}(0)=n ! /(n-k)$ ! for $k=0,1, \ldots, n$. Therefore
$$(1+x)^n=\sum_{k=0}^n \frac{n !}{k !(n-k) !} x^k$$

## 数学代写|实分析作业代写Real analysis代考|The Gamma Function

We close this chapter with a brief discussion of the Beta and Gamma functions, both of which are due to Euler. The Gamma function is closely related to factorials and arises in many areas of mathematics. The origin, history, and the development of the Gamma function are described very nicely in the article by Philip Davis listed in the supplemental reading. Our primary application of the Gamma function will be in the Taylor expansion of $(1-x)^{-\alpha}$, where $\alpha>0$ is arbitrary.

DEFINITION 8.8.1 For $00$, was given as Exercise 9 in Section 6.4. The Graph of $\Gamma(x)$ for $0<x<5$ is given in Figure 8.5. The following properties of the Gamma function show that it is closely related to factorials.
THEOREM 8.8.2 (a) For each $x, 0<x<\infty, \Gamma(x+1)=x \Gamma(x)$.
(b) For $n \in \mathbb{N}, \quad \Gamma(n+1)=n$ !.
Proof. Let $0<c<R<\infty$. We apply integration by parts to
$$\int_c^R t^x e^{-t} d t$$
With $u=t^x$ and $v^{\prime}=e^{-t}$,
\begin{aligned} \int_c^R t^x e^{-t} d t & =-\left.t^x e^{-t}\right|_c ^R+x \int_c^R t^{x-1} e^{-t} d t \ & =-\frac{R^x}{e^R}+c^x e^{-c}+x \int_c^R t^{x-1} e^{-t} d t \end{aligned}

# 实分析代考

## 数学代写|实分析作业代写Real analysis代考|Cauchy’s Form for the Remainder

$$R_n(x)=R_n(f, c)(x)=\frac{f^{(n+1)}(\zeta)}{n !}(x-c)(x-\zeta)^n$$

$$\int_c^x f^{(n+1)}(t)(x-t)^n d t=(x-c) f^{(n+1)}(\zeta)(x-\zeta)^n$$

$$f(x)=\sum_{k=0}^n \frac{f^{(k)}(0)}{k !} x^k$$

$$(1+x)^n=\sum_{k=0}^n \frac{n !}{k !(n-k) !} x^k$$

## 数学代写|实分析作业代写Real analysis代考|The Gamma Function

Gamma 函数与阶乘密切相关，出现在许多数学领域。Gamma 函数的起源、历史和发 展在补充阅读中列出的 Philip Davis 的文章中有很好的描述。Gamma 函数的主要应用 是在泰勒展开式中 $(1-x)^{-\alpha}$ ，在哪里 $\alpha>0$ 是任意的。

(b) 对于 $n \in \mathbb{N}, \quad \Gamma(n+1)=n !$ 。

$$\int_c^R t^x e^{-t} d t$$

$$\int_c^R t^x e^{-t} d t=-\left.t^x e^{-t}\right|_c ^R+x \int_c^R t^{x-1} e^{-t} d t \quad=-\frac{R^x}{e^R}+c^x e^{-c}+x \int_c^R t^{x-1} e^{-t} d t$$

myassignments-help数学代考价格说明

1、客户需提供物理代考的网址，相关账户，以及课程名称，Textbook等相关资料~客服会根据作业数量和持续时间给您定价~使收费透明，让您清楚的知道您的钱花在什么地方。

2、数学代写一般每篇报价约为600—1000rmb，费用根据持续时间、周作业量、成绩要求有所浮动(持续时间越长约便宜、周作业量越多约贵、成绩要求越高越贵)，报价后价格觉得合适，可以先付一周的款，我们帮你试做，满意后再继续，遇到Fail全额退款。

3、myassignments-help公司所有MATH作业代写服务支持付半款，全款，周付款，周付款一方面方便大家查阅自己的分数，一方面也方便大家资金周转，注意:每周固定周一时先预付下周的定金，不付定金不予继续做。物理代写一次性付清打9.5折。

Math作业代写、数学代写常见问题

myassignments-help擅长领域包含但不是全部: