Intervals of Convergence
In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?
∑ (from n = 0 to ∞) [ n xⁿ / (4ⁿ (n² + 1)) ]

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Identify the general term of the series: \(a_n = \frac{n x^n}{4^n (n^2 + 1)}\).

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

\[L = \lim_{n \to \infty} \left| \frac{(n+1) x^{n+1}}{4^{n+1} ((n+1)^2 + 1)} \cdot \frac{4^n (n^2 + 1)}{n x^n} \right| = \lim_{n \to \infty} \left| \frac{n+1}{n} \cdot \frac{n^2 + 1}{(n+1)^2 + 1} \cdot \frac{|x|}{4} \right|.\]

Simplify the limit by analyzing the behavior of the rational expressions as \(n \to \infty\), which will approach 1, so \(L = \frac{|x|}{4}\).

Set the condition for convergence from the Ratio Test: \(L < 1\), which gives \(\frac{|x|}{4} < 1\), or \(|x| < 4\). This means the radius of convergence is \(R = 4\) and the interval of convergence is initially \((-4, 4)\).

Check the endpoints \(x = -4\) and \(x = 4\) by substituting into the original series and testing for convergence (using appropriate tests such as the Alternating Series Test or p-series test) to determine if the series converges absolutely, conditionally, or diverges at these points.

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Radius and Interval of Convergence

The radius of convergence is the distance from the center of a power series within which the series converges. The interval of convergence includes all x-values for which the series converges, possibly including endpoints. Finding these involves applying tests like the Ratio or Root Test to determine where the series converges absolutely.

Absolute Convergence

A series converges absolutely if the series of absolute values converges. This means ∑|a_n| converges, ensuring the original series converges regardless of term signs. Absolute convergence implies stronger convergence and often simplifies analysis of power series.

Conditional Convergence

Conditional convergence occurs when a series converges, but not absolutely; that is, the series ∑a_n converges while ∑|a_n| diverges. This typically happens at the endpoints of the interval of convergence and requires careful testing, such as the Alternating Series Test, to confirm.

Intervals of ConvergenceIn Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?∑ (from n = 0 to ∞) [ n xⁿ / (4ⁿ (n² + 1)) ]