Combining power series Use the power series representationf(x ) =ln (1 − x) = −∑ₖ₌₁∞ xᵏ/k, for −1 ≤ x \u003c 1,to find the power series for the following functions (centered at 0). Give the interval of convergence of the new series.f(3x) = ln (1 − 3x)

Ch. 11 - Power Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Briggs 3rd Edition

Ch. 11 - Power Series

문제 11.2.47

11장, 문제 11.2.47

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.

f(3x) = ln (1 − 3x)

검증된 단계별 안내

Recall the given power series representation for the function $f(x) = \ln(1 - x)$, which is $f(x) = -\sum_{k=1}^{\infty} \frac{x^k}{k}$, valid for $-1 \leq x < 1$.

To find the power series for $f(3x) = \ln(1 - 3x)$, substitute \$3x$ in place of $x$ in the original series. This gives $f(3x) = -\sum_{k=1}^{\infty} \frac{(3x)^k}{k}$.

Rewrite the series by expanding the term $(3x)^k$ as \$3^k x^k$, so the series becomes $f(3x) = -\sum_{k=1}^{\infty} \frac{3^k x^k}{k}$.

Determine the interval of convergence for the new series by considering the original interval $-1 \leq x < 1$ and substituting \$3x$ for $x$. This means the inequality $|3x| < 1$ must hold, which simplifies to $|x| < \frac{1}{3}$.

Therefore, the power series for $f(3x)$ is $-\sum_{k=1}^{\infty} \frac{3^k x^k}{k}$, and its interval of convergence is $\left(-\frac{1}{3}, \frac{1}{3}\right)$.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Power Series Representation

A power series expresses a function as an infinite sum of terms involving powers of a variable, typically centered at a point (often zero). For example, ln(1 - x) can be written as -∑(x^k / k) for k=1 to ∞, valid within its radius of convergence. Understanding this allows manipulation and substitution within the series.

Substitution in Power Series

Substitution involves replacing the variable in a power series with another expression, such as replacing x by 3x. This changes each term's power accordingly and affects the interval of convergence. Correct substitution is essential to find the new series representation of functions like f(3x).

Interval of Convergence

The interval of convergence is the set of x-values for which a power series converges to the function. When substituting variables, the interval changes based on the new variable's domain. Determining this interval ensures the validity of the power series representation for the modified function.

Combining power series Use the power series representationf(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.f(3x) = ln (1 − 3x)