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 …

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 …

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 …

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 …

Aug 13, 2024 · In this section we will discuss in greater detail the convergence and divergence of infinite series. We will illustrate how partial sums are used to determine if an infinite series …

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.

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Standard examples: When using comparison tests, these are the things we are most likely to compare to: The geometric series $\sum c r^n$ converges absolutely if $\lvert r\rvert<1$ and …