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Suppose f(x) is continuous at a stationary point x_0. 1. If f^'(x)>0 on an open interval extending left from x_0 and f^'(x)<0 on an open interval extending right from x_0, ...

If u_n>0 and given B(n) a bounded function of n as n->infty, express the ratio of successive terms as |(u_n)/(u_(n+1))|=1+h/n+(B(n))/(n^r) for r>1. The series converges for ...

If, in an interval of x, sum_(r=1)^(n)a_r(x) is uniformly bounded with respect to n and x, and {v_r} is a sequence of positive non-increasing quantities tending to zero, then ...

Also known as the Leibniz criterion. An alternating series converges if a_1>=a_2>=... and lim_(k->infty)a_k=0.

Let suma_k and sumb_k be two series with positive terms and suppose lim_(k->infty)(a_k)/(b_k)=rho. If rho is finite and rho>0, then the two series both converge or diverge.

Let N be an odd integer, and assume there exists a Lucas sequence {U_n} with associated Sylvester cyclotomic numbers {Q_n} such that there is an n>sqrt(N) (with n and N ...

Let the probabilities of various classes in a distribution be p_1, p_2, ..., p_k, with observed frequencies m_1, m_2, ..., m_k. The quantity ...

Let {a_n} be a series of positive terms with a_(n+1)<=a_n. Then sum_(n=1)^(infty)a_n converges iff sum_(k=0)^infty2^ka_(2^k) converges.

D = f_(xx)f_(yy)-f_(xy)f_(yx) (1) = f_(xx)f_(yy)-f_(xy)^2, (2) where f_(ij) are partial derivatives.

A test to determine if a given series converges or diverges.