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# 数学代写|傅里叶分析代写Fourier Analysis代考|Special Series Expansions

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## 数学代写|傅里叶分析代写Fourier Analysis代考|Special Series Expansions

EXAMPLES OF INFINITE SERIES ARE geometric series, power series, and binomial series. These are discussed more fully in Sections $1.5,1.10$, and 1.11 , respectively. These series are defined as follows:

1. The sum of the geometric series exists for $|r|<1$ and is given by
$$\sum \mathrm{n}=0 \infty \mathrm{oc} \mathrm{arn}=\mathrm{a} 1-\mathrm{r}, \mathrm{rr}<1 .(1.16)$$
2. A power series expansion about $x=a$ with coefficient sequence $c_n$ is given by $\sum \mathrm{n}=0 \infty \mathrm{cn}(\mathrm{x}-\mathrm{a}) \mathrm{n}$
Taylor series expansion. Note that we will use $\sim$ to indicate that the series representation may not converge to $f(x)$ for all $x$. We will use equality when a convergence interval is indicated.
3. A Taylor series expansion of $f(x)$ about $x=a$ is the series
$$f(x) \sim \sum n=0 \infty c n(x-a) n,(1.17)$$
where
$$\mathrm{cn}=\mathrm{f}(\mathrm{n})(a) n ! .(1.18)$$
Maclaurin series expansion.
4. A Maclaurin series expansion of $f(x)$ is a Taylor series expansion of $f(x)$ about $x=0$, or
$\mathrm{f}(\mathrm{x})-\sum \mathrm{n}=0 \infty \mathrm{cnxn},(1.19)$
where
$$\mathrm{cn}=\mathrm{f}(\mathrm{n})(0) \mathrm{n} ! .(1.20)$$
Some common expansions are provided in Table 1.1.

## 数学代写|傅里叶分析代写Fourier Analysis代考|Power Series

Actually, what are now known as Taylor and Maclaurin series were known long before they were named. James Gregory (1638-1675) has been recognized for discovering Taylor series, which were later named after Brook Taylor (1685-1731). Similarly, Colin Maclaurin (1698-1746) did not actually discover Maclaurin series, but the name was adopted because of his particular use of series.

ANOTHER EXAMPLE OF AN INFINITE SERIES that the student has encountered in previous courses is the power series. Examples of such series are provided by Taylor and Maclaurin series.
A power series expansion about $x=a$ with coefficient sequence $c_n$ is given by $\sum \mathrm{n}=0 \infty \mathrm{cn}(\mathrm{x}-\mathrm{a}) \mathrm{n}$. For now we will consider all constants to be real numbers with $x$ in some subset of the set of real numbers.
Consider the following expansion about $x=0$ :
$$\sum \mathrm{n}=0 \infty \mathrm{xn}=1+\mathrm{x}+\mathrm{x} 2+\ldots .(1.23)$$
We would like to make sense of such expansions. For what values of $x$ will this infinite series converge? Until now we did not pay much attention to which infinite series might converge. However, this particular series is already familiar to us. It is a geometric series. Note that each term is gotten from the previous one through multiplication by $r=x$. The first term is $a=1$. So, from Equation (1.10), we have that the sum of the series is given by
$$\sum \mathrm{n}=0 \infty \mathrm{xn}=11-\mathrm{x}, \mathrm{x} \mid<1$$
In this case we see that the sum, when it exists, is a simple function. In fact, when $x$ is small, we can use this infinite series to provide approximations to the function $(1-x)^{-1}$. If $x$ is small enough, we can write
$$(1-x)-1 \approx 1+x$$
In Figure 1.16a we see that for small values of $x$ these functions do agree.

## 数学代写|傅里叶分析代写Fourier Analysis代考|Special Series Expansions

1. 几何级数的总和存在于 $|r|<1$ 并由
$$\sum \mathrm{n}=0 \infty \text { ocarn }=\mathrm{a} 1-\mathrm{r}, \mathrm{rr}<1$$
2. 关于幂级数展开 $x=a$ 带系数序列 $c_n$ 是 (谁) 给的 $\sum \mathrm{n}=0 \infty \mathrm{cn}(\mathrm{x}-\mathrm{a}) \mathrm{n}$ 泰勒级数展开。请注意，我们将使用 表明级数表示可能不会收敛到 $f(x)$ 对全部 $x$. 当指示收 敛区间时，我们将使用等式。
3. 的泰勒级数展开 $f(x)$ 关于 $x=a$ 是系列
$$f(x) \sim \sum n=0 \infty c n(x-a) n,(1.17)$$
在哪里
$$\mathrm{cn}=\mathrm{f}(\mathrm{n})(a) n ! .(1.18)$$
责克劳林级数展开。
4. 麦克劳林级数展开 $f(x)$ 是泰勒级数展开 $f(x)$ 关于 $x=0$ ， 或者 $f(x)-\sum n=0 \infty \mathrm{cnxn},(1.19)$
在哪里
$$\mathrm{cn}=\mathrm{f}(\mathrm{n})(0) \mathrm{n} ! .(1.20)$$
表 1.1 中提供了一些常见的扩展。

## 数学代写|傅里叶分析代写Fourier Analysis代考|Power Series

$$\sum \mathrm{n}=0 \infty \mathrm{xn}=1+\mathrm{x}+\mathrm{x} 2+\ldots$$

$$\sum n=0 \infty x n=11-x, x \mid<1$$

$$(1-x)-1 \approx 1+x$$

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