I just discovered that there is a convergence test called Raabe test

If $\displaystyle \lim_{n \to \infty}\inf \left\{ n \left(\frac{a_n}{a_{n+1}}-1 \right) \right\} >1 $ then the series $ \sum_{k=1 }^ \infty a_k$ is convergent and if$\displaystyle \lim_{n \to \infty}\sup \left\{ n \left(\frac{a_n}{a_{n+1}}-1 \right)\right\} <1$ then the series $ \sum_{k=1 }^ \infty a_k$ is divergent

I want to ask which book contains this theorem? It looks like this theorem is not in my book (Introduction to Real Analysis by Robert G. Bartle)


1 Answer 1


Along with the already mentioned Robert G. Bartle, Donald R. Sherbert - Introduction to Real Analysis, Fourth Edition -John Wiley & Sons, 2011, page 274 Raabe test is in following books:

  1. John M.H. Olmsted - Advanced calculus-Prentice Hall, 1961, page 396

  2. G.M. Fikhtengol'ts - Differential and Integral Calculus - vol. II, 2003, page 300

  3. Petrovic, John Srdjan - Advanced Calculus - Theory and Practice-CRC Press, 2013, page 491

  4. Angus E. Taylor, W. Robert Mann - Advanced Calculus-Wiley, 1991, page 591


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