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This lemma and the following proposition are due to Niels Henrik Abel (1802  1829).
Lemma: Abel's Lemma for Testing Convergence
The infinite series $\sum_{n=1}^\infty a_nb_n$ is convergent, if (by setting $A_0:=0$, $A_k:=\sum_{i=1}^k a_i$):
Table of Contents
Proofs: 1
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Proofs: 1 2
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References
Bibliography
 Heuser Harro: "Lehrbuch der Analysis, Teil 1", B.G. Teubner Stuttgart, 1994, 11th Edition