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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 11.2.49
Capitolo 11, Problema 11.2.49

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)

Guida verificata passo dopo passo
1
Recall the given power series for \( f(x) = \ln(1 - x) \): \[ f(x) = \ln(1 - x) = -\sum_{k=1}^{\infty} \frac{x^k}{k}, \quad \text{for } -1 \leq x < 1. \]
To find the power series for \( p(x) = 2x^6 \ln(1 - x) \), multiply the entire series for \( \ln(1 - x) \) by \( 2x^6 \): \[ p(x) = 2x^6 \cdot \left(-\sum_{k=1}^{\infty} \frac{x^k}{k}\right) = -2x^6 \sum_{k=1}^{\infty} \frac{x^k}{k}. \]
Combine the powers of \( x \) inside the summation: \[ p(x) = -2 \sum_{k=1}^{\infty} \frac{x^{k+6}}{k}. \]
Rewrite the series with a new index if desired (for example, let \( n = k + 6 \)) to express the series in standard power series form centered at 0: \[ p(x) = -2 \sum_{n=7}^{\infty} \frac{x^n}{n - 6}. \]
Determine the interval of convergence. Since the original series for \( \ln(1 - x) \) converges for \( -1 \leq x < 1 \), and multiplication by \( 2x^6 \) does not affect the radius of convergence, the interval of convergence for \( p(x) \) remains \( -1 \leq x < 1 \).

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Power Series Representation

A power series expresses a function as an infinite sum of terms involving powers of x, typically centered at a point (here, 0). Understanding how to write and manipulate these series is essential for representing functions like ln(1 − x) as sums that converge within a specific interval.
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Intro to Power Series

Interval of Convergence

The interval of convergence is the set of x-values for which a power series converges to the function it represents. Determining this interval ensures the validity of the series representation and is crucial when modifying or combining series, such as multiplying by 2x⁶.
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08:44
Interval of Convergence

Operations on Power Series

Power series can be added, multiplied, or composed by manipulating their coefficients and powers of x. For example, multiplying ln(1 − x) by 2x⁶ involves shifting the powers and scaling coefficients, which requires careful term-by-term adjustment to find the new series.
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Intro to Power Series
Pratica correlata
Domanda del libro di testo

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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ƒ(x) = (1 + x)^(1/3)"

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ƒ(x) = (1 + 2x)^(-5)

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Radius of convergence Find the radius of convergence for the following power series.

∑ₖ₌₁∞ (k!xᵏ)/(kᵏ)

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Approximating ln 2 Consider the following three ways to approximate

ln 2.

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