Final answer:
The question discusses the Comparison Test used in Mathematics to determine if a series converges or diverges by comparing it to another series with known convergence. The series ∑1/(n²−7) is compared to the known convergent series ∑1/n², and from the test, we conclude that the original series converges.
Step-by-step explanation:
Understanding the Comparison Test for Series
The question relates to the use of the Comparison Test for determining the convergence or divergence of a series. The Comparison Test involves comparing a series of interest with another series whose convergence is known. In the example given, we look at the series ∑1/(n²−7) and compare it to the convergent series ∑1/n². Since for all n>2, 1/(n²−7) is less than 1/n² and the series ∑1/n² is known to converge, by the Comparison Test, the series ∑1/(n²−7) also converges.
To apply the Comparison Test correctly, the terms of the series we are comparing must be positive beyond some point, and the series we compare to must be known to be convergent or divergent. If the terms of the series we are studying are always less than or equal to the terms of a convergent comparison series, then our series also converges. Conversely, if they are always greater than or equal to the terms of a divergent comparison series, then our series also diverges. This problem demonstrates proper application of the Comparison Test for convergence.