Let G be a graph on n vertices, and let lambda(1), lambda(2), ... ,lambda(n) be its eigenvalues. The Estrada index of G is a recently introduced molecular structure descriptor, defined as EE(G) = Sigma(n)(i=1)e. We show that the Estrada indices of the n-vertex cycle C-n and the n-vertex path P-n can be approximated as EE(C-n) approximate to n I-0 and EE(P-n) approximate to (n + 1)I-0 - cosh(2), where I-0 = Sigma(k >= 0)1(k!)(2) = 2.27958530.... The precision of these approximations is remarkably good. (c) 2007 Elsevier B.V. All rights reserved.