Textbook Question
Evaluating an infinite series Write the Maclaurin series for f(x) = ln (1+x) and find the interval of convergence. Evaluate f(−1/2) to find the value of ∑ₖ₌₁∞ 1/(k 2ᵏ)
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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 11.4.47
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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Recall the Taylor series expansion of the cosine function centered at 0 (Maclaurin series):
\[\cos x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots\]
To approximate \( \cos 2 \), substitute \( x = 2 \) into the series expansion:
\[\cos 2 = 1 - \frac{2^2}{2!} + \frac{2^4}{4!} - \frac{2^6}{6!} + \cdots\]
Identify the first four nonzero terms of this series explicitly:
1st term: \( 1 \)
2nd term: \( - \frac{2^2}{2!} \)
3rd term: \( + \frac{2^4}{4!} \)
4th term: \( - \frac{2^6}{6!} \)
Write the partial sum of these four terms as the approximation for \( \cos 2 \):
\[\cos 2 \approx 1 - \frac{2^2}{2!} + \frac{2^4}{4!} - \frac{2^6}{6!}\]
This expression represents the first four nonzero terms of the infinite Taylor series for \( \cos 2 \). You can stop here or calculate each factorial and power if a numerical approximation is desired.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Taylor Series Expansion
A Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point, usually around zero (Maclaurin series). It allows approximation of functions like cosine by polynomials, making complex functions easier to analyze and compute.
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Taylor Series
Maclaurin Series for Cosine
The Maclaurin series for cosine is a specific Taylor series centered at zero, given by cos(x) = 1 - x²/2! + x⁴/4! - x⁶/6! + ... . This alternating series includes only even powers of x and is used to approximate cosine values near zero.
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Term Selection and Approximation Accuracy
When approximating functions using Taylor series, selecting the first few nonzero terms balances simplicity and accuracy. Including more terms improves precision but increases complexity. For cos(2), using the first four nonzero terms provides a good polynomial approximation of the value.
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Related Practice
Textbook Question
Binomial series Write out the first three terms of the Maclaurin series for the following functions.
ƒ(x) = (1 + 2x)^(-5)
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Textbook Question
Combining power series Use the geometric series
f(x) = 1/(1-x) = ∑ₖ₌₀∞ xᵏ, for |x| < 1,
to find the power series representation for the following functions (centered at 0). Give the interval of convergence of the new series.
g(x) = x³/(1 − x)
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Textbook Question
Differentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series.
g(x) = 2/(1 − 2x)² using f(x) = 1/(1 − 2x)
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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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