Textbook Question
Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers.
cos 2
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Ch. 11 - Power Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- 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
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 11, Problem 11.R.46
Binomial series Write out the first three terms of the Maclaurin series for the following functions.
ƒ(x) = (1 + 2x)^(-5)
Verified step by step guidance1
Recall that the Maclaurin series is a special case of the Taylor series expanded at \(x = 0\). For a function \(f(x)\), the Maclaurin series is given by \(f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n\).
Recognize that the function \(f(x) = (1 + 2x)^{-5}\) can be expanded using the binomial series formula for negative integer exponents: \( (1 + u)^k = \sum_{n=0}^{\infty} \binom{k}{n} u^n \), where \(k = -5\) and \(u = 2x\).
Write the general term of the binomial series for this function as \( \binom{-5}{n} (2x)^n \), where the binomial coefficient for negative integers is defined as \( \binom{k}{n} = \frac{k (k-1) (k-2) \cdots (k - n + 1)}{n!} \).
Calculate the first three terms by substituting \(n = 0, 1, 2\) into the general term:
- For \(n=0\): \(\binom{-5}{0} (2x)^0\)
- For \(n=1\): \(\binom{-5}{1} (2x)^1\)
- For \(n=2\): \(\binom{-5}{2} (2x)^2\).
Simplify each term by evaluating the binomial coefficients and powers of \$2x$ to write out the first three terms of the Maclaurin series explicitly.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Maclaurin Series
The Maclaurin series is a special case of the Taylor series expanded at x = 0. It represents a function as an infinite sum of terms calculated from the derivatives of the function at zero. Writing out the first few terms involves evaluating the function and its derivatives at zero and expressing them as powers of x.
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Convergence of Taylor & Maclaurin Series
Binomial Series Expansion
The binomial series generalizes the binomial theorem to any real exponent, allowing expansion of expressions like (1 + x)^n where n can be negative or fractional. It expresses the function as an infinite sum involving binomial coefficients, which are calculated using combinations or the generalized formula for non-integer powers.
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06:00Geometric Series
Calculating Binomial Coefficients for Negative Powers
For negative integer exponents, binomial coefficients are computed using the formula involving factorials or the generalized binomial coefficient definition. These coefficients determine the sign and magnitude of each term in the series, crucial for correctly expanding functions like (1 + 2x)^(-5).
Recommended video:
04:09The Power Rule: Negative & Rational Exponents
Related Practice
Textbook Question
Write out the first three terms of the Maclaurin series for the following functions.
ƒ(x) = (1 + x)^(1/3)"
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Textbook Question
Radius of convergence Find the radius of convergence for the following power series.
∑ₖ₌₁∞ (k!xᵏ)/(kᵏ)
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Textbook Question
Combining power series Use the power series representation
f(x ) =ln (1 − x) = −∑ₖ₌₁∞ xᵏ/k, for −1 ≤ x < 1,
to find the power series for the following functions (centered at 0). Give the interval of convergence of the new series.
p(x) = 2x⁶ ln(1 − x)
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Textbook Question
Taylor polynomials Find the nth-order Taylor polynomial for the following functions centered at the given point a.
ƒ(x) = cos⁻¹ x, n = 2, a = 1/2
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Textbook Question
Approximating ln 2 Consider the following three ways to approximate
ln 2.
a. Use the Taylor series for ln (1 + x) centered at 0 and evaluate it at x = 1 (convergence was asserted in Table 11.5). Write the resulting infinite series.
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