Finding Taylor or Maclaurin Series for a Function · 求函数的泰勒或麦克劳林级数
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| Taylor series/ˈteɪlə ˈsɪəriːz/ | 泰勒级数 | tài lēi jí shù |
| Maclaurin series/məˈklɔːrɪn ˈsɪəriːz/ | 麦克劳林级数 | mài kè láo lín jí shù |
The polynomial that never stops
- A Taylor polynomial stops at degree $n$. Let it run forever and you get a Taylor series 泰勒级数.
- $\displaystyle\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^{n}$ — the infinite version, exactly representing $f$ where it converges.
- Centered at $0$, it's a Maclaurin series 麦克劳林级数.
- A handful of these are worth knowing by heart.
永不停止的多项式
- 泰勒多项式止步于 $n$ 次。让它永远跑下去,你就得到泰勒级数。
- $\displaystyle\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^{n}$——无穷版本,在它收敛之处精确表示 $f$。
- 以 $0$ 为中心,它是麦克劳林级数。
- 少数几个值得烂熟于心。
A Taylor series centered at $0$ is called a ____ series. · 以 $0$ 为中心的泰勒级数称为 ____ 级数。
Maclaurin = Taylor centered at $0$. · 麦克劳林 = 以 $0$ 为中心的泰勒级数。
The must-know Maclaurin series
- $\displaystyle e^{x}=\sum_{n=0}^{\infty}\frac{x^{n}}{n!}=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots$
- $\displaystyle \sin x=\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{(2n+1)!}=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$
- $\displaystyle \cos x=\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{(2n)!}=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots$
- $\displaystyle \frac{1}{1-x}=\sum_{n=0}^{\infty}x^{n}=1+x+x^2+\cdots$ (the geometric series, $|x|<1$).
必背的麦克劳林级数
- $\displaystyle e^{x}=\sum_{n=0}^{\infty}\frac{x^{n}}{n!}=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots$
- $\displaystyle \sin x=\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{(2n+1)!}=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$
- $\displaystyle \cos x=\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{(2n)!}=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots$
- $\displaystyle \frac{1}{1-x}=\sum_{n=0}^{\infty}x^{n}=1+x+x^2+\cdots$(几何级数,$|x|<1$)。
cos x from its series · 从级数推导 cos x
$\cos x=1-\tfrac{x^2}{2!}+\tfrac{x^4}{4!}-\cdots$ — even powers, alternating signs, factorial denominators. · $\cos x=1-\tfrac{x^2}{2!}+\tfrac{x^4}{4!}-\cdots$ — 偶次幂，交替符号，阶乘分母。
The Maclaurin series for $e^x$ is... · $e^x$ 的麦克劳林级数是...
All powers, coefficients $\tfrac1{n!}$, no alternation. · 所有幂次，系数 $\tfrac1{n!}$，无交替。
The series for $\dfrac{1}{1-x}$ (with $|x|<1$) is... · $\dfrac{1}{1-x}$ (带 $|x|<1$) 的级数是...
The geometric series $1+x+x^2+\cdots$. · 几何级数 $1+x+x^2+\cdots$。
Building one from scratch
- Compute $f(a),f'(a),f''(a),\dots$ and plug into $\frac{f^{(n)}(a)}{n!}(x-a)^n$, term by term.
- Look for the pattern in the derivatives to write the general $n$th term.
- The known series above are usually derived once then reused — memorize them.
- The general term is what makes it a series, not just a few terms.
从头构造一个
- 计算 $f(a),f'(a),f''(a),\dots$,逐项代入 $\frac{f^{(n)}(a)}{n!}(x-a)^n$。
- 在导数中找规律,写出一般的第 $n$ 项。
- 上面已知的级数通常推导一次后重用——把它们背下来。
- 一般项才使它成为级数,而非几项。
Using $\cos x\approx1-\tfrac{x^2}{2}$, estimate $\cos(0.2)$. · 使用 $\cos x\approx1-\tfrac{x^2}{2}$ 估计 $\cos(0.2)$。
$1-\tfrac{0.04}{2}=0.98$.
$\sin$ and $\cos$ share a family
- Notice $\sin x$ uses odd powers and $\cos x$ uses even powers — and both alternate sign.
- $e^x$ has every power with all-positive coefficients $\tfrac1{n!}$.
- These patterns let you write out terms quickly and spot which series a problem wants.
- Recognizing the family is half the battle on the exam.
$\sin$ 与 $\cos$ 同一家族
- 注意 $\sin x$ 用奇次幂,$\cos x$ 用偶次幂——且两者都交替变号。
- $e^x$ 有每个幂,系数全为正的 $\tfrac1{n!}$。
- 这些规律让你快速写出各项,并看出题目要哪个级数。
- 认出家族在考试中成功了一半。
The Maclaurin series for $\sin x$ uses... · $\sin x$ 的麦克劳林级数使用...
$\sin x=x-\tfrac{x^3}{3!}+\cdots$ (odd, alternating). · $\sin x=x-\tfrac{x^3}{3!}+\cdots$ (奇次，交替)。
The denominators in these standard series are factorials. · 这些标准级数的分母是阶乘。
The $n$th term has $n!$ (or $(2n)!$, $(2n+1)!$). · 第 $n$ 项含有 $n!$ (或 $(2n)!$, $(2n+1)!$)。
Keep the factorials and the sign pattern straight: $\sin x$ has odd powers $x^{2n+1}$ (starts with $x$), $\cos x$ has even powers $x^{2n}$ (starts with $1$), both alternating; $e^x$ has all powers, no alternation. Mixing up odd/even or dropping the $(-1)^n$ is the classic slip.
把阶乘和符号规律分清:$\sin x$ 是奇次幂 $x^{2n+1}$(从 $x$ 开始),$\cos x$ 是偶次幂 $x^{2n}$(从 $1$ 开始),两者都交替;$e^x$ 有所有幂,无交替。搞混奇/偶或丢掉 $(-1)^n$ 是经典失误。
Write the Maclaurin series for $\cos x$ and use it to approximate $\cos(0.2)$ to two terms.
- $\cos x=1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}-\cdots$
- Two terms at $x=0.2$: $1-\dfrac{0.04}{2}=1-0.02=0.98$.
- True $\cos 0.2\approx0.980$. ✓
写出 $\cos x$ 的麦克劳林级数,并用它以两项近似 $\cos(0.2)$。
- $\cos x=1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}-\cdots$
- 在 $x=0.2$ 的两项:$1-\dfrac{0.04}{2}=1-0.02=0.98$。
- 真值 $\cos 0.2\approx0.980$。✓
A Taylor series is the infinite $\sum\frac{f^{(n)}(a)}{n!}(x-a)^n$; centered at $0$ it's a Maclaurin series. Memorize $e^x$, $\sin x$, $\cos x$, and $\frac{1}{1-x}$. Watch the factorials and sign patterns: $\sin$ = odd powers, $\cos$ = even powers, both alternating; $e^x$ = all powers, no alternation.
泰勒级数是无穷的 $\sum\frac{f^{(n)}(a)}{n!}(x-a)^n$;以 $0$ 为中心即麦克劳林级数。背下 $e^x$、$\sin x$、$\cos x$ 与 $\frac{1}{1-x}$。注意阶乘与符号规律:$\sin$ = 奇次幂,$\cos$ = 偶次幂,两者交替;$e^x$ = 所有幂,无交替。