The Derivation of the Exponential Map of Matrices
Gijs M. Tuynman · American Mathematical Monthly · 1995
recapitulates this proof in [1], seems to assume that for a sequence hk that goes to O the quotients hm/htl for m > n are bounded. And as a last remark: Dini's test for divergence says that Eak is divergent if there exists a (weakly) monotone increasing sequence ck = mkak with El/Mk (£ak/ck) diverging. Ae monotonicit condition can be written as aJak+l 2 (l/mk)/(l/mk+l) and by a standard argument that makes Eak diverge also. (Ihat is Dini's proof.) Let us here replace the monotonicity of the ck by the weaker condition that the ck are bounded below by a positive number, say by 1. Then we have ak 2 ak/ck, with the series of the latter terms diverging by assumption, and thus a slightly improved version of Dini's criterion also turns out to be identical with the comparison test (divergence version)! The interesting aspect is again that writing the ck as mkak makes it easy to set up specific tests