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Let p run over all distinct primitive ordered periodic geodesics, and let tau(p) denote the positive length of p, then every even function h(rho) analytic in ...

The sine function sinx is one of the basic functions encountered in trigonometry (the others being the cosecant, cosine, cotangent, secant, and tangent). Let theta be an ...

Suppose the harmonic series converges to h: sum_(k=1)^infty1/k=h. Then rearranging the terms in the sum gives h-1=h, which is a contradiction.

Given a geometric sequence {a_1,a_1r,a_1r^2,...}, the number r is called the common ratio associated to the sequence.

The series sumf(n) for a monotonic nonincreasing f(x) is convergent if lim_(x->infty)^_(e^xf(e^x))/(f(x))<1 and divergent if lim_(x->infty)__(e^xf(e^x))/(f(x))>1.

A function whose value decreases more quickly than any polynomial is said to be an exponentially decreasing function. The prototypical example is the function e^(-x), plotted ...

A function whose value increases more quickly than any polynomial is said to be an exponentially increasing function. The prototypical example is the function e^x, plotted ...

Given a hypergeometric series sum_(k)c_k, c_k is called a hypergeometric term (Koepf 1998, p. 12).

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.

Little-omega notation is the inverse of the Landau symbol o, i.e., f(n) in o(phi(n)) <==> phi(n) in omega(f(n)).