Mathematics

Mathematics

Problem

Which of the following statements is NOT a necessary hypothesis for applying the integral test to determine the convergence of the series ∑n=1∞an\sum_{n=1}^{\infty} a_n? a) The sequence {an} is eventually positive.\text{The sequence }\{a_n\}\text{ is eventually positive.} b) The function that matches an is continuous on [1,∞).\text{The function that matches }a_n\text{ is continuous on }[1,\infty). c) The function that matches an is bounded above on [1,∞).\text{The function that matches }a_n\text{ is bounded above on }[1,\infty). d) The function that matches an is eventually decreasing.\text{The function that matches }a_n\text{ is eventually decreasing.}
Intuitive explanation

The integral test requires that the function f(x)f(x) corresponding to an=f(n)a_n = f(n) satisfy specific properties—but not all conditions are equally essential. The key is recognizing which properties are truly necessary for the test's logic to work: you need ff to be positive and decreasing so that rectangles under the curve reliably bound the series terms. One condition is often listed but turns out to be redundant or weaker than required.

Intuitive explanation

The integral test requires that the function f(x)f(x) corresponding to an=f(n)a_n = f(n) satisfy specific properties—but not all conditions are equally essential. The key is recognizing which properties are truly necessary for the test's logic to work: you need ff to be positive and decreasing so that rectangles under the curve reliably bound the series terms. One condition is often listed but turns out to be redundant or weaker than required.

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