Math for Non-Geeks/ Direct comparison test

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In this chapter we will study an important convergence criterion called the direct comparison test. This criterion allows us to deduce the convergence behavior of a series by comparing it to that of another series. That way we can answer questions regarding the convergence of a series by considering a simpler series. Using this criterion we can estimate upper or lower bounds of the series, until we hopefully find a proof for the convergence behavior.

The direct comparison test is also used to proof other criteria such as the quotient test and the root test, which are both useful tools to have when solving problems about the convergence of series.

Direct comparison test: majorant

There are two versions of the direct comparison test. The first one is a test for convergence and involves finding a majorant of the series:

Direct comparison test: minorant

The second comparison test is similar, but we use it to determine the divergence of a series, and it involves finding a minorant of that series.

Examples and exercises

Direct comparison with minorant

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Corollary: Limit comparison test

For series with positive summands there is another test called the limit comparison test, which we can derive form the direct comparison test:

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