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What is the Comparison Test?

Comparison Test

The Comparison Test is a method used in calculus to determine the convergence or divergence of an infinite series by comparing its terms with those of another series with known convergence properties. The test involves comparing the given series with a known convergent or divergent series to establish its behavior.

Given an infinite series ∑a_n and a second series ∑b_n, the Comparison Test states:

  1. If 0 ≤ a_n ≤ b_n for all n, and ∑b_n converges, then ∑a_n also converges.
  2. If 0 ≤ b_n ≤ a_n for all n, and ∑b_n diverges, then ∑a_n also diverges.

The Comparison Test is particularly useful for series with non-negative terms and can be applied when the terms of the given series have a similar form to those of a known convergent or divergent series. It is often used in conjunction with other convergence tests, such as the Ratio Test or the Root Test, to determine the behavior of an infinite series.

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