Radius and interval of convergence Use the Ratio Test or the Root Test to determine the radius of convergence of the following power series. Test the endpoints to determine the interval of convergence, when appropriate.

∞
Σ (x - 1)ᵏ/(k5ᵏ)
k = 1

검증된 단계별 안내

Identify the general term of the power series: \(a_k = \frac{(x - 1)^k}{k 5^k}\).

Apply the Ratio Test, which involves computing the limit \(L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|\). Substitute \(a_k\) and \(a_{k+1}\) into this expression.

Simplify the ratio inside the limit: \(\left| \frac{(x - 1)^{k+1}}{(k+1) 5^{k+1}} \cdot \frac{k 5^k}{(x - 1)^k} \right| = \left| \frac{x - 1}{5} \cdot \frac{k}{k+1} \right|\).

Evaluate the limit as \(k\) approaches infinity: \(L = \left| \frac{x - 1}{5} \right| \cdot \lim_{k \to \infty} \frac{k}{k+1} = \left| \frac{x - 1}{5} \right|\).

Set the limit \(L\) less than 1 to find the radius of convergence: \(\left| \frac{x - 1}{5} \right| < 1\). Solve this inequality to find the interval of convergence before testing the endpoints.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Radius of Convergence

The radius of convergence of a power series is the distance from the center of the series within which the series converges absolutely. It defines an interval on the x-axis where the series behaves well. Finding this radius helps determine where the series can be used to represent a function accurately.

Ratio Test and Root Test

The Ratio Test and Root Test are methods to determine the convergence of infinite series. The Ratio Test examines the limit of the ratio of successive terms, while the Root Test looks at the nth root of the absolute value of terms. Both tests help find the radius of convergence for power series by analyzing term behavior as k approaches infinity.

Interval of Convergence and Endpoint Testing

The interval of convergence is the full set of x-values for which a power series converges. After finding the radius, endpoints must be tested separately because convergence at these points is not guaranteed. Testing endpoints involves substituting them into the series and checking for convergence using appropriate tests.

Radius and interval of convergence Use the Ratio Test or the Root Test to determine the radius of convergence of the following power series. Test the endpoints to determine the interval of convergence, when appropriate.∞Σ (x - 1)ᵏ/(k5ᵏ)k = 1