In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series .

Sep 23, 2025 · In mathematics, convergence tests are essential tools used to determine whether an infinite series converges (approaches a finite value) or diverges (grows without bound or oscillates).

You can view the transcript for this segmented clip of “Mixed Convergence Tests” here (opens in new window). In the following table, we summarize the convergence tests and when each can be applied.

Ratio Test If the limit of | a [n +1]/ a [n]| is less than 1, then the series (absolutely) converges. If the limit is larger than one, or infinite, then the series diverges.

Testing for Convergence or Divergence of a Series Many of the series you come across will fall into one of several basic types. Recognizing these types will help you decide which tests or strategies will be …

Jan 22, 2022 · For each pair of series below, decide whether the second series is a valid comparison series to determine the convergence of the first series, using the direct comparison test and/or the …

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Comparison test: This applies only to positive series. If [Math Processing Error] 0 ≤ a n ≤ b n and [Math Processing Error] ∑ b n converges, then [Math Processing Error] ∑ a n converges.

In this situation, one can often determine whether a given series converges or diverges without explicitly calculating lim k → ∞ s k k→∞lim sk, via one of the following tests for convergence.

As we have seen, it is often easier to determine the convergence of a sequence than finding (meaning recognizing a nice expression) its limit. Likewise, it is often easier to determine the convergence of a …