Problem 47 - Essential Calculus

Question

Give examples that show that the convergence of a power series at an endpoint of its interval of convergence may be either conditional or absolute. Explain your reasoning.

Step-by-Step Solution

Answer

The alternating harmonic series \(Σ((-1)^n/n)\) provides an example of conditional convergence at an end point, because it converges, but does not converge when all terms are replaced by their absolute values. On the other hand, the series \(Σx^n/n!\) is an example of absolute convergence because it does converge when all terms are replaced by their absolute values.

Step 1: Provide an Example of Conditional Convergence

Consider the alternating harmonic series \(-1^n/n\) for \(x = -1\), also known as \(Σ((-1)^n/n)\). This series converges conditionally at \(x = -1\) because its terms do not converge to 0.

Step 2: Explain Convergence Test

The alternating series test states that if the terms decrease to zero, which they do in the above series as \(n\) approaches infinity, the series converges.

Step 3: Provide an Example of Absolute Convergence

Now consider the series \(Σx^n/n!\) for \(x = -1\) which equals \(Σ((-1)^n/n!)\). This series also converges at \(x = -1\) but unlike the above series, it converges absolutely.

Step 4: Explain Absolute Convergence Test

The ratio test of convergence states if the absolute value of the ratio of the \(n+1\)th term to the nth term is less than 1, the series converges absolutely. This holds true for this series.