For us to be able to apply the integral comparison test to a function [math: f(x)], this function has to have how many of the following properties?a) [math: f(x)] has to be non-negativeb) [math: f(x)] has to be decreasing c) [math: f(x)] has to integrabled) [math: limx→∞f(x)=0]\lim_{x\to \infty} f(x)=0e) [math: f(x)] has to be increasingf) [math: f(x)] has to be differentiable简答题
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Question

For us to be able to apply the integral comparison test to a function [math: f(x)], this function has to have how many of the following properties?a) [math: f(x)] has to be non-negativeb) [math: f(x)] has to be decreasing c) [math: f(x)] has to integrabled) [math: limx→∞f(x)=0]\lim_{x\to \infty} f(x)=0e) [math: f(x)] has to be increasingf) [math: f(x)] has to be differentiable简答题

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BlackTom AI · Accepted Answer

To determine what properties are required to apply the integral comparison test to a function f(x), we need to recall the standard setup of the test and what it relies on.
Option a) f(x) has to be non-negative: This is typically required because the integral test compares a positive term series to an improper integral. If f takes negative values, the direct comparison to a positive series becomes invalid, so this condition is necessary.
Option b) f(x) has to be decreasing: Monotonic decrease ensures that the tails of the series and the integral can be bounded in a consistent way, which is essential for the comparison argument. Without monotonic decrease, the neat correspondence between partial sums and the integral breaks down, so this is another needed property.
Option c) f(x) has to be integrable: Specifically, the improper integral ∫ from some point to ∞ f(x) dx must exist (converge or diverge) to relate it to the series. If the integral isn’t even defined (not integrable in the improper sense), the test cannot be applied, so integrability is a required ingredient.
Option d) lim_{x→∞} f(x) = 0: This limit is a consequence if the integral converges, but it is not a required hypothesis to apply the test. The integral test can be discussed in terms of positivity, monotonicity, and improper integrability without asserting the limit to zero a priori.
Option e) f(x) has to be increasing: This is contrary to the standard needs of the test. Increasing functions do not provide the monotone, tail-bounding behavior the integral test relies on, so this property is not only unnecessary but typically incorrect for the test.
Option f) f(x) has to be differentiable: Differentiability is not part of the classic hypotheses of the integral test. The test usually only requires positivity, monotonicity, and integrability; differentiability is a stronger condition that isn’t required here.

Putting these together, the essential properties are: a) non-negative, b) decreasing, and c) integrable. The remaining properties (d, e, f) are not required and in some cases are not compatible with the standard use of the test.

Therefore, the number of properties required for applying the integral comparison test among the listed options is 3 (a, b, and c).

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