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Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 11 - Power SeriesProblema 11.3.35
Capitolo 11, Problema 11.3.35

Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.


ln (1 + x²)

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Recall the Taylor series expansion for \( \ln(1+x) \) centered at 0, which is given by: \[ \ln(1+x) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \] This is the series you will manipulate to find the series for \( \ln(1+x^2) \).
Substitute \( x^2 \) in place of \( x \) in the series for \( \ln(1+x) \). This gives: \[ \ln(1+x^2) = x^2 - \frac{(x^2)^2}{2} + \frac{(x^2)^3}{3} - \frac{(x^2)^4}{4} + \cdots \] which simplifies to: \[ \ln(1+x^2) = x^2 - \frac{x^4}{2} + \frac{x^6}{3} - \frac{x^8}{4} + \cdots \]
Identify the first four nonzero terms from the series after substitution. These terms correspond to the powers of \( x \) with nonzero coefficients in the expansion.
Write out explicitly the first four nonzero terms of the Taylor series for \( \ln(1+x^2) \) centered at 0, using the simplified powers and coefficients from the previous step.
Verify that the series is centered at 0 and that the terms are ordered by increasing powers of \( x \). This confirms the correctness of the Taylor series expansion for \( \ln(1+x^2) \).

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Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives at a single point, usually centered at zero (Maclaurin series). It approximates functions locally and is useful for expressing complex functions as polynomials.
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Taylor Series

Manipulating Known Series

To find the Taylor series of a new function, you can use known series expansions and apply algebraic operations like substitution, multiplication, or composition. For example, substituting x² into the series for ln(1 + x) helps find the series for ln(1 + x²).
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Geometric Series

Identifying Nonzero Terms

When expanding a series, it is important to identify and list the first few nonzero terms, as some terms may vanish due to the function's structure or substitution. This ensures the approximation captures the function's behavior accurately near the center.
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Divergence Test (nth Term Test)
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Remainders Find the remainder in the Taylor series centered at the point a for the following functions. Then show that lim ₙ→∞ Rₙ(x)=0, for all x in the interval of convergence.


f(x) = e⁻ˣ, a = 0

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Radius and interval of convergence Determine the radius and interval of convergence of the following power series.


∑ₖ₌₀∞ (2x)ᵏ

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Probability: tossing for a head The expected (average) number of tosses of a fair coin required to obtain the first head is ∑ₖ₌₁∞ k(1/2)ᵏ. Evaluate this series and determine the expected number of tosses. (Hint: Differentiate a geometric series.) 

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Any method


a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients.


b. Determine the radius of convergence of the series.


f(x) = (1 + x²)⁻²/³

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Radius and interval of convergence Determine the radius and interval of convergence of the following power series.


−x²/1 + x⁴/2! −x⁶/3! + x⁸/4! − ⋯

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Use of Tech Linear and quadratic approximation


a. Find the linear approximating polynomial for the following functions centered at the given point a.


b. Find the quadratic approximating polynomial for the following functions centered at a.


c Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.


f(x) = cos x, a = π/4; approximate cos (0.24π)

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