Convergence tests

In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series ∑ n = 1 ∞ a n {displaystyle sum _{n=1}^{infty }a_{n}} . In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series ∑ n = 1 ∞ a n {displaystyle sum _{n=1}^{infty }a_{n}} . If the limit of the summand is undefined or nonzero, that is lim n → ∞ a n ≠ 0 {displaystyle lim _{n o infty }a_{n} eq 0} , then the series must diverge. In this sense, the partial sums are Cauchy only if this limit exists and is equal to zero. The test is inconclusive if the limit of the summand is zero. This is also known as D'Alembert's criterion. This is also known as the nth root test or Cauchy's criterion. The root test is stronger than the ratio test: whenever the ratio test determines the convergence or divergence of an infinite series, the root test does too, but not conversely.