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 x⁴ᵏ/k²
k = 1

Verified step by step guidance

1. Identify the general term of the power series: \(a_k = \frac{x^{4k}}{k^2}\).
2. Apply the Ratio Test, which involves computing the limit \(L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|\).
3. Substitute \(a_k\) and \(a_{k+1}\) into the ratio: \(\left| \frac{a_{k+1}}{a_k} \right| = \left| \frac{x^{4(k+1)}}{(k+1)^2} \cdot \frac{k^2}{x^{4k}} \right| = \left| x^4 \right| \cdot \frac{k^2}{(k+1)^2}\).
4. Evaluate the limit as \(k\) approaches infinity: \(L = |x|^4 \cdot \lim_{k \to \infty} \frac{k^2}{(k+1)^2} = |x|^4\).
5. Use the Ratio Test criterion for convergence: the series converges if \(L < 1\), so \(|x|^4 < 1\), which implies \(|x| < 1\). This gives the radius of convergence \(R = 1\). Next, test the endpoints \(x = -1\) and \(x = 1\) by substituting into the original series and checking for convergence.

Key Concepts

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 is found by analyzing the limit behavior of the series' terms, often using tests like the Ratio or Root Test. This radius defines an interval on the real line where the series converges.

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 absolute value of consecutive term ratios, 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 evaluating limits involving the variable.

Interval of Convergence and Endpoint Testing

The interval of convergence is the set of all x-values for which a power series converges. After finding the radius of convergence, 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 convergence tests.