교과서 질문
Taylor series
b. Write the power series using summation notation.
f(x) = 1/x, a = 1
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Ch. 11 - Power Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
11장, 문제 11.3.21b
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x)=3ˣ, a=0
검증된 단계별 안내1
Recall that the Taylor series of a function \( f(x) \) centered at \( a = 0 \) (also called the Maclaurin series) is given by the formula:
\[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n \]
where \( f^{(n)}(0) \) is the \( n \)-th derivative of \( f(x) \) evaluated at \( x=0 \).
Identify the function given: \( f(x) = 3^x \). To find the Taylor series, we need to compute the derivatives \( f^{(n)}(x) \) and then evaluate them at \( x=0 \).
Calculate the first few derivatives of \( f(x) = 3^x \):
- \( f(x) = 3^x \)
- \( f'(x) = 3^x \ln(3) \)
- \( f''(x) = 3^x (\ln(3))^2 \)
- \( f^{(n)}(x) = 3^x (\ln(3))^n \)
This pattern holds for all \( n \geq 0 \).
Evaluate the \( n \)-th derivative at \( x=0 \):
\[ f^{(n)}(0) = 3^0 (\ln(3))^n = (\ln(3))^n \]
Substitute \( f^{(n)}(0) \) into the Taylor series formula to write the power series in summation notation:
\[ f(x) = \sum_{n=0}^{\infty} \frac{(\ln(3))^n}{n!} x^n \]
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
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Taylor Series Expansion
A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point, called the center (a). For f(x) = 3^x at a = 0, the series uses derivatives evaluated at 0 to approximate the function near that point.
추천 영상:
08:42
Taylor Series
Power Series and Summation Notation
A power series is an infinite sum of terms in the form c_n(x - a)^n, where c_n are coefficients and a is the center. Summation notation compactly expresses this series as Σ c_n (x - a)^n, making it easier to write and analyze the series.
추천 영상:
05:58Intro to Power Series
Interval of Convergence
The interval of convergence is the set of x-values for which the power series converges to the function. Determining this interval ensures the series accurately represents the function within that range, often found using ratio or root tests.
추천 영상:
08:44Interval of Convergence
관련 실천
교과서 질문
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x) = (1 + x²)⁻¹, a = 0
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교과서 질문
{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals
S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt
b. Expand sin t² and cos t² in a Maclaurin series, and then integrate to find the first four nonzero terms of the Maclaurin series for S and C.
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교과서 질문
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x) = e⁻ˣ, a=0
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교과서 질문
{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero.
a. Estimate f(0.1) and give a bound on the error in the approximation.
f(x) = tan⁻¹ x ≈ x
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교과서 질문
Taylor series
b. Write the power series using summation notation.
f(x) = ln x, a = 3
110
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