Radius and interval of convergence Determine the radius and interval of convergence of the following power series.

∑ₖ₌₁∞ sinᵏ(1/k) xᵏ

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

To find the radius of convergence, use the root test which involves calculating \(\limsup_{k \to \infty} \sqrt[k]{|a_k|} = \limsup_{k \to \infty} \sqrt[k]{\left|\sin^k\left(\frac{1}{k}\right) x^k\right|}\).

Simplify the expression inside the limit: \(\sqrt[k]{|a_k|} = \left|\sin\left(\frac{1}{k}\right)\right| \cdot |x|\).

Evaluate the limit \(\lim_{k \to \infty} \sin\left(\frac{1}{k}\right)\) by using the fact that \(\sin y \approx y\) for small \(y\), so \(\sin\left(\frac{1}{k}\right) \approx \frac{1}{k}\), which tends to 0 as \(k \to \infty\).

Since the limit of \(\sin\left(\frac{1}{k}\right)\) is 0, the root test limit becomes \(0 \cdot |x| = 0\) for all \(x\), indicating the radius of convergence is infinite. Then, verify the interval of convergence by checking the behavior of the series at all real \(x\) values.

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주요 개념

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Power Series and Convergence

A power series is an infinite sum of terms in the form a_k x^k, where a_k depends on k. Understanding convergence means determining for which values of x the series sums to a finite value. This involves analyzing the behavior of the coefficients and the variable x.

Radius of Convergence

The radius of convergence is the distance from the center of the power series (usually zero) within which the series converges absolutely. It can be found using tests like the root or ratio test applied to the coefficients a_k. The radius defines an interval on the x-axis where the series behaves well.

Interval of Convergence

The interval of convergence includes all x-values for which the power series converges, typically from -R to R, where R is the radius of convergence. Endpoints must be checked separately since convergence there is not guaranteed. This interval determines the domain where the series represents a valid function.

Radius and interval of convergence Determine the radius and interval of convergence of the following power series.∑ₖ₌₁∞ sinᵏ(1/k) xᵏ