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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 $A_0:=0$, $A_k:=\sum_{i=1}^k a_i$):

• the series $\sum_{n=0}^\infty A_k(b_k-b_{k+1})$ is a convergent series and
• the sequence $(A_n b_{n+1})_{n\in\mathbb N}$ is convergent sequence.

Table of Contents

Proofs: 1

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Proofs: 1 2

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References

Bibliography

1. Heuser Harro: "Lehrbuch der Analysis, Teil 1", B.G. Teubner Stuttgart, 1994, 11th Edition