Textbook Question
Limits by power series Use Taylor series to evaluate the following limits.
lim ₙ → ₄ ln (x - 3)/(x² - 16)
Ch. 11 - Power Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
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.37
Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.
ƒ(x) = cos x, a = π/2
Verified step by step guidance1
Identify the function and the center point: here, the function is \(f(x) = \cos x\) and the center is \(a = \frac{\pi}{2}\).
Recall the Taylor series formula centered at \(a\):
\[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n,\]
where \(f^{(n)}(a)\) is the \(n\)th derivative of \(f\) evaluated at \(x = a\).
Calculate the first few derivatives of \(f(x) = \cos x\) and evaluate them at \(x = \frac{\pi}{2}\):
- \(f(x) = \cos x\) so \(f\left(\frac{\pi}{2}\right) = 0\)
- \(f'(x) = -\sin x\) so \(f'\left(\frac{\pi}{2}\right) = -1\)
- \(f''(x) = -\cos x\) so \(f''\left(\frac{\pi}{2}\right) = 0\)
- \(f'''(x) = \sin x\) so \(f'''\left(\frac{\pi}{2}\right) = 1\)
Write out the first three nonzero terms of the Taylor series using the derivatives found:
\[f(x) \approx f(a) + \frac{f'(a)}{1!}(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3 + \cdots\]
Substitute the values and keep only the first three nonzero terms.
Express the Taylor series in summation notation:
\[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}\left(\frac{\pi}{2}\right)}{n!} (x - \frac{\pi}{2})^n,\]
where you can identify the pattern of nonzero terms from the derivatives calculated.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
6mWas this helpful?
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 derivatives of the function at a single point. Each term involves the nth derivative evaluated at the center point, multiplied by (x - a)^n and divided by n!. This allows approximation of functions near the point a.
Recommended video:
08:42
Taylor Series
Derivatives of Trigonometric Functions
Understanding the derivatives of cosine is essential, as they follow a repeating cycle: cos x, -sin x, -cos x, sin x, then back to cos x. Evaluating these derivatives at the center point helps determine the coefficients of the Taylor series terms.
Recommended video:
06:35Derivatives of Other Inverse Trigonometric Functions
Summation Notation for Series
Summation notation concisely expresses infinite or finite sums using the sigma symbol (∑). Writing the Taylor series in summation form involves identifying the general term formula, including factorial denominators and powers of (x - a), to represent the series compactly.
Recommended video:
06:00Geometric Series
Related Practice
Textbook Question
Taylor polynomials Find the nth-order Taylor polynomial for the following functions centered at the given point a.
ƒ(x) = sinh (-3x), n = 3, a = 0
Textbook Question
Power series from the geometric series Use the geometric series a Σₖ ₌ ₀ ∞ (x)ᵏ = 1/(1 - x), for |x| < 1, to determine the Maclaurin series and the interval of convergence for the following functions.
ƒ(x) = ln (1 - 4x)
Textbook Question
Taylor polynomials Find the nth-order Taylor polynomial for the following functions centered at the given point a.
ƒ(x) = sin 2x, n = 3, a = 0
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. There is more than one way to choose the center of the series.
sin 20°
Textbook Question
Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.
ƒ(x) = 1/(4 + x²), a = 0