Convergence Tests: The Comparison Test for Series

When dealing with the convergence of series, mathematicians often rely on various tests to determine whether a series converges or diverges. One of the most fundamental and versatile tests is the Comparison Test. This test provides a powerful tool for analyzing the behavior of complex series by comparing them to simpler, well-known series whose convergence properties are already understood.

The Basics of the Comparison Test

Consider two series of non-negative terms:

The Comparison Test states that if the series ∑ bn converges and an ≤ bn eventually, then ∑ an also converges.

Conversely, if ∑ an diverges and an ≤ bn eventually, then ∑ bn also diverges.

This test is particularly useful for situations where direct computation of convergence is difficult, but comparison with a simpler series can clarify the behavior of the original series.

Practical Examples

Example 1: A Convergent Series

Suppose we want to determine the convergence of the series ∑ sin(n)/n!.

We know from our knowledge of series that:

∑n0∞ 1/n! e, which converges.

By the Comparison Test, since |sin(n)/n!| ≤ 1/n! for all n, and the series ∑ 1/n! converges, it follows that the series ∑ sin(n)/n! also converges.

Example 2: A Divergent Series

Consider the series ∑ 1/n. This is a well-known harmonic series, which is known to diverge.

Suppose we have another series ∑ an such that an ≤ 1/n eventually.

By the Comparison Test, since ∑ 1/n diverges, and an ≤ 1/n for sufficiently large n, it follows that ∑ an also diverges.

General Usage and Limitations

The Comparison Test is a fundamental tool in the study of series convergence. However, it is important to note that the test only provides information when the compared series is known to be convergent or divergent. If the compared series is itself unknown or uncertain, the test may not be directly applicable.

Moreover, the Comparison Test can sometimes be subtle in its application. For instance, if the terms of the series are not non-negative, other tests such as the Integral Test or the Limit Comparison Test may be required.

Conclusion

The Comparison Test is a crucial and versatile tool in the analysis of series convergence. By comparing a given series with a known series, it provides a powerful method to determine convergence or divergence without needing to perform detailed computations.

For mathematicians and students, mastering the application of the Comparison Test is essential for tackling complex problems in real analysis and advanced calculus.