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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 11.3.37
Capitolo 11, Problema 11.3.37

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.


1/(1 − 2x)

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1
Recall the Taylor series expansion for the function \( \frac{1}{1 - x} \) centered at 0, which is given by the geometric series: \( \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \cdots \).
To find the Taylor series for \( \frac{1}{1 - 2x} \), substitute \( 2x \) in place of \( x \) in the geometric series expansion. This gives: \( \frac{1}{1 - 2x} = \sum_{n=0}^{\infty} (2x)^n = 1 + 2x + (2x)^2 + (2x)^3 + \cdots \).
Expand the powers of \( 2x \) in each term to write the series explicitly: \( 1 + 2x + 4x^2 + 8x^3 + \cdots \).
Identify the first four nonzero terms of the series, which correspond to the terms with powers \( x^0, x^1, x^2, \) and \( x^3 \).
Write the first four nonzero terms as \( 1 + 2x + 4x^2 + 8x^3 \), which is the Taylor series approximation of \( \frac{1}{1 - 2x} \) centered at 0 up to the third degree.

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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, usually centered at zero (Maclaurin series). It approximates functions using polynomials, making complex functions easier to analyze and compute.
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Taylor Series

Geometric Series and Its Taylor Series

The geometric series 1/(1 - x) can be expanded as 1 + x + x^2 + x^3 + ... for |x| < 1. Recognizing this form helps in manipulating similar functions like 1/(1 - 2x) by substituting variables, enabling straightforward derivation of their Taylor series.
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Term-by-Term Substitution and Manipulation

To find the Taylor series of functions like 1/(1 - 2x), substitute the variable inside the known series (e.g., replace x with 2x in the geometric series). Then, expand and simplify term-by-term to identify the first four nonzero terms of the series centered at zero.
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Percorso guidato
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Divergence Test (nth Term Test)
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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.


sinh x²

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Differential equations


a. Find a power series for the solution of the following differential equations, subject to the given initial condition

b. Identify the function represented by the power series.


y′(t) − y = 0, y(0) = 2

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Remainders Find the remainder Rₙ for the nth−order Taylor polynomial centered at a for the given functions. Express the result for a general value of n.


f(x) = 1/(1 - x), a=0

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Limits Evaluate the following limits using Taylor series.

lim ₓ→∞ x sin(1/x)

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

∑ₖ₌₀∞ (k²⁰ xᵏ)/(2k+1)!

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In terms of the remainder, what does it mean for a Taylor series for a function f to converge to f?

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