Pronunciation: [ɹˈɑːbz tˈɛst] (IPA) 
The correct spelling of the word "Raabes test" is /ˈʁaːbəs tɛst/. The initial "R" is pronounced as a voiced uvular fricative /ʁ/ in German. The "a" is pronounced as a long "a" sound /aː/ followed by a short "e" sound /ə/. The final "s" is pronounced as an unvoiced "s" sound /st/. Raabes test is a medical test used to assess the ability of the stomach to secrete hydrochloric acid.
Raabe's test, also known as Raabe's formula, is a mathematical test used to determine the convergence or divergence of a given series. It is named after the German mathematician, Friedrich Raabe, who developed this test.
Raabe's test is specifically applied to series with positive terms. It is most commonly used when the series does not fall into the criteria of other convergence tests, such as the ratio test or the root test. The main advantage of Raabe's test is that it allows us to check the convergence using only a few terms of the given series.
The test involves evaluating the limit of the sequence:
Ln = n × ((an / an+1) - 1)
Where "an" represents the terms of the given series. If the resulting limit is greater than 1, then the series is convergent. Conversely, if the limit is less than 1, the series is divergent. If the limit equals 1, the test is inconclusive.
Raabe's test is generally preferred when the ratio test or root test yields no definite conclusion. However, it is important to note that Raabe's test may not always provide a conclusive result for all series. In such cases, other convergence tests may be required for a definitive judgment on the convergence or divergence of the series.
The term "Raabe's test" is named after the German mathematician Johann Ludwig Raabe. The test, also known as Raabe's ratio test, is a convergence test used in mathematics to determine the convergence or divergence of a series. Johann Ludwig Raabe formulated this test in the 19th century, and it subsequently became popular in mathematical analysis.
