Taylor series and interval of convergence

b. Write the power series using summation notation.

f(x) = ln (x − 2), a = 3

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Identify the function and the center of the Taylor series expansion. Here, the function is \(f(x) = \ln(x - 2)\) and the center is \(a = 3\).

Rewrite the function in terms of \((x - a)\) to express it as a power series centered at \(x = 3\). Set \(u = x - 3\), so that \(x - 2 = (x - 3) + 1 = u + 1\).

Recall the Taylor series expansion for \(\ln(1 + u)\) around \(u = 0\), which is \(\ln(1 + u) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{u^n}{n}\) for \(|u| < 1\).

Substitute back \(u = x - 3\) into the series to write \(f(x)\) as a power series centered at \(x = 3\): \(f(x) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{(x - 3)^n}{n}\).

Express the final answer in summation notation: \(f(x) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{(x - 3)^n}{n}\), which represents the power series expansion of \(\ln(x - 2)\) about \(x = 3\).

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주요 개념

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

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 a. It approximates the function near that point using powers of (x - a), allowing complex functions to be expressed as polynomials.

Power Series and Summation Notation

A power series is an infinite series of the form Σ c_n (x - a)^n, where c_n are coefficients and a is the center. Summation notation concisely expresses this infinite sum, making it easier to manipulate and analyze the series.

Interval of Convergence

The interval of convergence is the set of x-values for which a power series converges to the function. Determining this interval involves testing the radius of convergence and checking endpoints to ensure the series accurately represents the function within that range.

Taylor series and interval of convergenceb. Write the power series using summation notation.f(x) = ln (x − 2), a = 3