Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. Suppose f is a continuous, positive, decreasing function, for x ≥ 1, and aₖ = f(k), for k = 1, 2, 3, …. If ∑ (k = 1 to ∞) aₖ converges to L, then ∫ (1 to ∞) f(x) dx converges to L.

Accepted Answer

Step 1: Understand the problem statement. We have a function $$f$$ that is continuous, positive, and decreasing for $$x \geq 1$$, and a sequence $$a_k = f(k)$$ for $$k = 1, 2, 3, \ldots. $$The series $$\sum_{k=1}^\infty a_k$$ converges to some limit $$L. We $$need to determine if the improper integral $$\$$int_1$$^\infty f(x) \, dx$$ also converges to the same limit $$L. $$Step 2: Recall the Integral Test for convergence of series. The Integral Test states that if $$f is $$continuous, positive, and decreasing on $$[1, \infty)$$, then the series $$\sum_{k=1}^\infty f(k)$$ and the integral $$\$$int_1$$^\infty f(x) \, dx$$ either both converge or both diverge. However, this test does not say that their sums or values are equal, only that their convergence behavior matches. Step 3: Analyze the relationship between the sum and the integral. The sum $$S = \sum_{k=1}^\infty f(k) is a $$discrete sum of function values at integers, while the integral $$I = \$$int_1$$^\infty f(x) \, dx is $$the continuous area under the curve. Although both converge or diverge together, their exact values are generally not equal. Step 4: Consider a counterexample to show the values are not necessarily equal. For example, take $$f(x) = \frac{1}{x^2$}. $$The series $$\sum_{k=1}^\infty \frac{1}{k^2$}$$ converges to $$\frac{\pi^2$}{6}$$, while the integral $$\$$int_1$$^\infty \frac{1}{x^2$} \, dx$$ converges to 1. Since $$\frac{\pi^2$}{6} 
eq 1$$, the integral and series limits are not equal. Step 5: Conclusion: The statement is false because although the series and integral both converge, the integral does not necessarily converge to the same limit $$L as $$the series. The Integral Test guarantees convergence behavior but not equality of sums and integrals.

Risposta video verificata per un problema simile:

Concetti chiave

Comparison between Series and Improper Integrals

Integral Test for Convergence

Counterexamples in Convergence Analysis