Geodesic
A Remark on Completely Monotonic Sequences, with an Application to Summability
Canadian mathematical bulletin, Tome 6 (1963) no. 2, pp. 171-173

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A sequence of non-negative numbers, μ0 μ1 ... μn, is called completely monotonic [5, p. 108] if (-1)n Δn μk ≧ 0 for n, k = 0, 1, 2, .... Such sequences occur in many connexions, such as the Hausdorff moment problem and Hausdorff summability [1, Chapter XI, 5, Chapters III and IV].It is natural to inquire as to the circumstances under which the inequality "≧" above can be strengthened to ">". As it happens, this can be done always, except for sequences all of whose terms past the first are identical. The formal statement follows. (In it, as above, Δnμk is the n-th forward difference, i. e. Δ0μk = μk; Δnμk = Δn-1 μk+1 - Δn-1μk.) 
Lorch, Lee; Moser, Leo. A Remark on Completely Monotonic Sequences, with an Application to Summability. Canadian mathematical bulletin, Tome 6 (1963) no. 2, pp. 171-173. doi: 10.4153/CMB-1963-016-3
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