How many of the following statements are true? a) The integral test remainder estimate gives us an estimate of how close a partial sum of a non-negative decreasing convergent series is to the true sum of that series.b) The integral test remainder estimate tells us the exact error we make when we truncate a suitable infinite series at a certain term.c) The integral test remainder estimate applies to all infinite series.d) The integral test remainder estimate also sometimes applies to divergent series.e) There is no such thing as the integral test remainder estimate. It should be integral remainder estimate.

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BlackTom AI · Accepted Answer

We’re asked to assess five statements about the integral test remainder estimate and determine how many of them are true. Below, I examine each claim in turn, then compare them to the standard facts about the integral test and its remainder estimate.

Option a) The integral test remainder estimate gives us an estimate of how close a partial sum of a non-negative decreasing convergent series is to the true sum of that series.
This is correct. For a series with positive terms that are decreasing, the integral test provides a way to bound the tail (the remainder) of the series by an integral, giving an estimate of the error in truncating after N terms. While it is not an exact value, it yields a bound and a sense of closeness between the partial sum and the limit when the series meets the hypotheses (nonnegative, decreasing, convergent).

Option b) The integral test remainder estimate tells us the exact error we make when we truncate a suitable infinite series at a certain term.
This is not accurate. The remainder estimate from the integral test gives bounds (an inequality) on the error, not the exact error in general. The exact error would require more delicate information or, in many cases, is not known. So this statement is false.

Option c) The integral test remainder estimate applies to all infinite series.
This is false. The integral test requires the terms a_n to be positive, decreasing, and relate to a continuous, positive decreasing function f with a_n = f(n). Not all infinite series satisfy these conditions (for example, series with non-monotone terms or negative terms do not fit the standard integral test), so the remainder estimate does not apply universally.

Option d) The integral test remainder estimate also sometimes applies to divergent series.
This is misleading. The integral test itself can be used to determine convergence or divergence of a series with positive decreasing terms by comparing to an improper integral. If the corresponding integral diverges, the series diverges; if the integral converges, the series converges. However, the remainder estimate (which quantifies error in truncation) is meaningful only for convergent series; for divergent series, there isn’t a finite sum to approximate, so speaking of a remainder estimate in the same sense is inappropriate. Therefore, this statement is false.

Option e) There is no such thing as the integral test remainder estimate. It should be integral remainder estimate.
This is incorrect. The standard terminology is indeed the “integral test remainder estimate” (or “remainder in the integral test”). The claim that there is no such thing is just a matter of naming, but it conflicts with accepted terminology and is not a mathematically correct description of the concept. So this statement is false.

In light of these evaluations, exactly one statement among a–e is true, namely option a. The remaining statements misstate the scope, precision, or applicability of the integral test remainder estimate or use incorrect terminology.

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