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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First, rewrite the given power series in a general term form. Notice the pattern of powers and factorials: the nth term looks like \((-1)^n \frac{x^{2(n+1)}}{(n+1)!}\), starting from \(n=0\).

Identify the general term \(a_n = (-1)^n \frac{x^{2(n+1)}}{(n+1)!}\). To analyze convergence, it is often easier to express the series as \(\sum_{n=0}^\infty (-1)^n \frac{x^{2(n+1)}}{(n+1)!}\).

Apply the Ratio Test to find the radius of convergence. Compute the limit \(L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\) where \(a_n = (-1)^n \frac{x^{2(n+1)}}{(n+1)!}\).

Simplify the ratio inside the limit: \(\left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{x^{2(n+2)}}{(n+2)!} \cdot \frac{(n+1)!}{x^{2(n+1)}} \right| = \left| \frac{x^2}{n+2} \right|\).

Evaluate the limit as \(n \to \infty\): \(L = \lim_{n \to \infty} \left| \frac{x^2}{n+2} \right| = 0\) for all real \(x\). Since \(L=0 < 1\) for all \(x\), the radius of convergence is infinite, meaning the series converges for all real \(x\). Thus, the interval of convergence is \((-\infty, \infty)\).

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주요 개념

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

Power Series

A power series is an infinite sum of terms in the form a_n(x - c)^n, where a_n are coefficients and c is the center. Understanding the structure of power series helps analyze their convergence behavior depending on the value of x.

Radius of Convergence

The radius of convergence is the distance from the center within which a power series converges absolutely. It can be found using tests like the Ratio Test or Root Test, indicating the interval where the series represents a valid function.

Interval of Convergence

The interval of convergence is the set of all x-values for which the power series converges. It includes the radius of convergence and requires checking endpoints separately to determine if the series converges or diverges there.

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! − ⋯