# 数学代写|实分析作业代写Real analysis代考|Taylor’s Theorem with Integral Remainder

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## 数学代写|实分析作业代写Real analysis代考|Taylor’s Theorem with Integral Remainder

There are several forms to the remainder term in the one-variable Taylor’s Theorem for real-valued functions, and the differences already show up in their lowest-order formulations. Let $f$ be given, and, for definiteness, suppose $a<x$. If $o(1)$ denotes a term that tends to 0 as $x$ tends to $a$, three such lowest-order formulas are
$$\begin{array}{ll} f(x)=f(a)+o(1) & \text { if } f \text { is merely assumed to be continuous, } \ f(x)=f(a)+(x-a) f^{\prime}(\xi) & \text { with } a<\xi<x \text { if } f \text { is continuous } \ & \text { on }[a, x] \text { and } f^{\prime} \text { exists on }(a, x), \ f(x)=f(a)+\int_a^x f^{\prime}(t) d t & \text { if } f \text { and } f^{\prime} \text { are continuous on }[a, x] . \end{array}$$
The first formula follows directly from the definition of continuity, while the second formula restates the Mean Value Theorem and the third formula restates part of the Fundamental Theorem of Calculus. The hypotheses of the three formulas increase in strength, and so do the conclusions. In practice, Taylor’s Theorem is most often used with functions having derivatives of all orders, and then the strongest hypothesis is satisfied. Thus we state a general theorem corresponding only to the third formula above. It applies to complex-valued functions as well as real-valued functions.

Theorem 1.36 (Taylor’s Theorem). Let $n$ be an integer $\geq 0$, let $a$ and $x$ be points of $\mathbb{R}$, and let $f$ be a complex-valued function with $n+1$ continuous derivatives on the closed interval from $a$ to $x$. Then
$$f(x)=f(a)+\frac{f^{\prime}(a)}{1 !}(x-a)+\cdots+\frac{f^{(n)}(a)}{n !}(x-a)^n+R_n(a, x),$$
where
$$R_n(a, x)= \begin{cases}\frac{1}{n !} \int_a^x(x-t)^n f^{(n+1)}(t) d t & \text { if } a \leq x \ -\frac{1}{n !} \int_x^a(x-t)^n f^{(n+1)}(t) d t & \text { if } x \leq a\end{cases}$$

## 数学代写|实分析作业代写Real analysis代考|Power Series and Special Functions

A power series is an infinite series of the form $\sum_{n=0}^{\infty} c_n z^n$. Normally in mathematics, if nothing is said to the contrary, the coefficients $c_n$ are assumed to be complex and the variable $z$ is allowed to be complex. However, in the context of real-variable theory, as when forming derivatives of functions defined on intervals, one is interested only in real values of $z$. In this book the context will generally make clear whether the variable is to be regarded as complex or as real.

One source of power series is the “infinite Taylor series” $\sum_{n=0}^{\infty} \frac{f^{(n)}(0) x^n}{n !}$ of a function $f$ having derivatives of all orders, with the remainder terms discarded. In this case the variable is to be real. If the series is convergent at $x$, the series has sum $f(x)$ if and only if $\lim _n R_n(0, x)=0$. Later in this section, we shall see examples where the limit is identically 0 and where it is nowhere 0 for $x \neq 0$.

Theorem 1.37. If a power series $\sum_{n=0}^{\infty} c_n z^n$ is convergent in $\mathbb{C}$ for some complex $z_0$ with $\left|z_0\right|=R$ and if $R^{\prime}R$, then $\sum_{n=0}^{\infty} c_n z_0^n$ diverges.

PROOF. The theorem is vacuous unless $R>0$. Since $\sum_{n=0}^{\infty} c_n z_0^n$ is convergent, the terms $c_n z_0^n$ tend to 0 . Thus there is some integer $N$ for which $\left|c_n\right| R^n \leq 1$ when $n \geq N$. Fix $R^{\prime}<R$. For $|z| \leq R^{\prime}$ and $n \geq N$, we have
$$\left|c_n z^n\right|=\left|c_n z_0^n\right|\left|\frac{z}{z_0}\right|^n=\left|c_n\right| R^n\left|\frac{z}{z_0}\right|^n \leq\left(\frac{R^{\prime}}{R}\right)^n .$$

# 实分析代写

## 数学代写|实分析作业代写Real analysis代考|Taylor’s Theorem with Integral Remainder

$$\begin{array}{ll} f(x)=f(a)+o(1) & \text { if } f \text { is merely assumed to be continuous, } \ f(x)=f(a)+(x-a) f^{\prime}(\xi) & \text { with } a<\xi<x \text { if } f \text { is continuous } \ & \text { on }[a, x] \text { and } f^{\prime} \text { exists on }(a, x), \ f(x)=f(a)+\int_a^x f^{\prime}(t) d t & \text { if } f \text { and } f^{\prime} \text { are continuous on }[a, x] . \end{array}$$

$$f(x)=f(a)+\frac{f^{\prime}(a)}{1 !}(x-a)+\cdots+\frac{f^{(n)}(a)}{n !}(x-a)^n+R_n(a, x),$$

$$R_n(a, x)= \begin{cases}\frac{1}{n !} \int_a^x(x-t)^n f^{(n+1)}(t) d t & \text { if } a \leq x \ -\frac{1}{n !} \int_x^a(x-t)^n f^{(n+1)}(t) d t & \text { if } x \leq a\end{cases}$$

## 数学代写|实分析作业代写Real analysis代考|Power Series and Special Functions

$$\left|c_n z^n\right|=\left|c_n z_0^n\right|\left|\frac{z}{z_0}\right|^n=\left|c_n\right| R^n\left|\frac{z}{z_0}\right|^n \leq\left(\frac{R^{\prime}}{R}\right)^n .$$

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