Finding Taylor or Maclaurin Series for a Function
Introduced| English | Meaning |
|---|---|
| Taylor series/ˈteɪlə ˈsɪəriːz/ | Taylor series |
| Maclaurin series/məˈklɔːrɪn ˈsɪəriːz/ | A Taylor series centred at zero, whose coefficients are determined by the function's derivatives at zero. |
| partial sum/ˈpɑːʃl sʌm/ | partial sum |
| remainder/rɪˈmeɪndə/ | remainder |
| factorial/fækˈtɔːrɪəl/ | The product of the positive integers up to a given positive integer; zero factorial is defined as one. |
| geometric series/ˌdʒiːəʊˈmetrɪk ˈsɪəriːz/ | An infinite series a + ar + ar² + … with constant ratio r; for nonzero a it converges to a/(1 − r) exactly when |r| < 1. |
Calculate cosine using powers
- A calculator can approximate $\cos(0.2)$ with $1-0.2^2/2=0.98$. A Taylor series 泰勒级数 explains where these terms come from.
- A finite sum gives an approximation. The infinite series gives an exact representation only when its limit equals the function.
Choose the center first
- At a center $x=a$, the Taylor series is $\sum_{n=0}^{\infty}\dfrac{f^{(n)}(a)}{n!}(x-a)^n$. Here $f^{(0)}=f$ and $0!=1$.
- At $a=0$, it is a Maclaurin series 麦克劳林级数. Evaluate every derivative at the center before writing the terms.
A Taylor series centered at 0 is called a ____ series.
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Maclaurin is the special case a=0.
What is the coefficient of $(x-a)^n$ in the Taylor series of f about a?
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Evaluate the nth derivative at a, then divide by n factorial.
The n=0 Taylor term is f(a), because 0!=1.
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The constant term reproduces the function value at the center.
Separate a partial sum from a limit
- The Taylor polynomial $P_N(x)=\sum_{n=0}^{N}\dfrac{f^{(n)}(a)}{n!}(x-a)^n$ is a partial sum 部分和. Its degree is at most $N$; some coefficients may be zero.
- The remainder 余项 is $R_N(x)=f(x)-P_N(x)$. To write $f(x)=\lim_{N\to\infty}P_N(x)$, you need $R_N(x)\to0$ at that $x$.
A series can converge without its sum equalling the original function. Do not use convergence alone to justify a Taylor representation. The four standard identities in this lesson have known validity intervals.
What condition justifies $f(x)=\lim_{N\to\infty}P_N(x)$ at a fixed x?
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The difference f(x)−P_N(x) must tend to zero.
If the Taylor series of a smooth function converges, convergence alone proves its sum equals the function.
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Equality also requires the remainder f−P_N to tend to zero.
Derive the cosine pattern
- The derivatives of $\cos x$ repeat: $\cos x$, $-\sin x$, $-\cos x$, $\sin x$. At $0$, their values are $1,0,-1,0$, then the pattern repeats.
- Divide by the factorial 阶乘 of the derivative order. This gives $1+0x-x^2/2!+0x^3+x^4/4!-\cdots$.
What is $f^{(4)}(0)$ for $f(x)=\cos x$?
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After four derivatives the function returns to $\cos x$, giving $\cos(0)=1$.
What is the coefficient of $x^4$ in the Maclaurin series of $\cos x$?
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Divide the fourth derivative value $1$ by $4!=24$.
Write the cosine general term
- The nonzero powers are $2n$, and their signs are $(-1)^n$. The matching denominator is $(2n)!$.
- Use $n=0,1,2$ to check the first three nonzero terms. The series represents $\cos x$ for every real $x$.
The first three nonzero cosine Maclaurin terms have powers 0,2 and ____.
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Cosine uses even powers 2n.
Sine uses the odd powers
- The sine derivative values at $0$ are $0,1,0,-1$, then repeat. Its first nonzero term is $x$.
- Use powers $2n+1$ and denominator $(2n+1)!$. This identity is valid for every real $x$, with angles in radians.
The nonzero terms in the Maclaurin series for $\sin x$ use...
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The powers are 1,3,5,… and the signs alternate.
What is the coefficient of $x^5$ in the Maclaurin series for $\sin x$?
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The third nonzero term is $+x^5/5!$, with $5!=120$.
For n=0, what does $(-1)^nx^{2n+1}/(2n+1)!$ become?
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The sign is positive, the exponent is $1$, and $1!=1$.
The exponential keeps every power
- Every derivative of $e^x$ equals $e^x$. At $0$, each derivative value is $1$.
- So $e^x=\sum_{n=0}^{\infty}x^n/n!=1+x+x^2/2!+x^3/3!+\cdots$, for every real $x$. There is no $(-1)^n$ factor.
Which is the Maclaurin series for $e^x$, starting at n=0?
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All derivatives at $0$ equal $1$, giving coefficient $1/n!$.
The standard Maclaurin series for $e^x$, $\sin x$ and $\cos x$ use factorial denominators.
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These three series use n!, (2n+1)! and (2n)!, respectively. The geometric-series coefficients are 1.
The geometric identity has a boundary
- The geometric series 几何级数 is $\dfrac{1}{1-x}=\sum_{n=0}^{\infty}x^n=1+x+x^2+\cdots$ for $|x|<1$. Its coefficients are all $1$.
- At $x=1$ and $x=-1$, its terms do not tend to $0$. Its full interval is $(-1,1)$, even though the function also exists at many points outside it.
Which series represents $1/(1-x)$ for $|x|<1$?
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Every coefficient is 1; the radius is 1 and both endpoints diverge.
Which standard Maclaurin series represent their functions for every real x?
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Sine, cosine and exponential have infinite radius; the geometric identity requires |x|<1.
Build a series away from zero
- For $f(x)=e^x$ centered at $a=1$, every derivative value is $e$. Each term uses $(x-1)^n$, not $x^n$.
- The result is valid for every real $x$. Changing the center changes the powers and their coefficients.
At $x=1$, all positive-power terms vanish and the constant term is $e$, as required.
The Taylor expansion uses powers $(x-1)^n$. What is its center?
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The center is the value that makes x−1 zero.
For $e^x$ centered at 1, what multiplies $(x-1)^2$?
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Every derivative at 1 equals e, then divide by 2!.
Count nonzero terms separately from degree
- For cosine, the first two nonzero terms give $1-x^2/2$. This has degree $2$, although it contains only two nonzero terms. Its degree-$1$ Taylor polynomial is just $1$.
- At $x=0.2$, $1-0.2^2/2=0.98$. Adding $0.2^4/24$ gives $0.9800666\ldots$. The corresponding exact value is about $0.98006658$.
Function and Maclaurin partial sums
The default two nonzero cosine terms give 1−x²/2, degree 2. Compare actual finite sums with their functions, including the geometric series outside its convergence interval.
Using the first two nonzero Maclaurin terms, $1-x^2/2$, estimate $\cos(0.2)$.
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$1-0.04/2=0.98$.
What is the degree of $1-x^2/2+x^4/24$?
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The largest power with a nonzero coefficient is 4, although there are three nonzero terms.
What is the degree of the polynomial made from the first three nonzero sine Maclaurin terms?
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The powers are 1,3 and 5. Three terms do not mean degree 3.
Try a different center and a derivative table
- Build the series for $\sin x$ centered at $\pi/2$. Let $h=x-\pi/2$ and use $\sin(\pi/2+h)=\cos(h)$.
- A table of finitely many derivative values gives a finite Taylor polynomial. Without a general pattern, it does not determine every term of an infinite series.
Find the Taylor series for $e^x$ centered at 2. Give its first three terms, general term and validity interval.
Use derivative values at 2 and powers of x−2.
Find the Taylor series for $\sin x$ centered at $\pi/2$. Use $h=x-\pi/2$ and $\sin(\pi/2+h)=\cos(h)$. Give three nonzero terms, general term and validity interval.
Use the cosine Maclaurin series in the shifted variable h.
A smooth function has $f(2)=3$, $f'(2)=-2$, $f''(2)=8$, $f^{(3)}(2)=0$, $f^{(4)}(2)=24$. Write its Taylor polynomial through order 4 about 2. Do these values determine its entire Taylor series?
Divide each derivative value by its order factorial; the higher derivatives are unknown.
Check the structure of the answer
- State the center, starting index, general term and validity interval. Check the constant term and the first few powers against derivative values.
- Distinguish the finite approximation from the exact infinite identity. Watch factorials, signs, zero coefficients and the difference between degree and term count.
Taylor coefficients are $f^{(n)}(a)/n!$; Maclaurin means $a=0$. The standard sine, cosine and exponential series represent their functions for all real $x$. The geometric identity requires $|x|<1$. Equality to a Taylor series requires the remainder to tend to zero.
Review: what positive integer is the denominator of the $x^6$ term in the cosine Maclaurin series?
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The term is $-x^6/6!$, and $6!=720$.