Proposition: Abel's Test

If the infinite series $\sum_{k=1}^\infty a_k$ is convergent and the sequence1 $(b_k)_{k\in\mathbb N}$ is bounded and monotonic, then the series $\sum_{k=1}^\infty a_kb_k$ is also convergent.

Proofs: 1


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References

Bibliography

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

Footnotes


  1. If you want to apply this convergence criterion to complexed-valued series $\sum_{k=1}^\infty a_kb_k$, you have to make sure that $(b_k)_{k\in\mathbb N}$ remains a real-valued sequence. Otherwise, the monotonicity criterion will lose its sense.