44. Assumptions Matter In 1829, Lejeune Dirichlet pointed out that the great French mathematician Augustin Louis Cauchy made a mistake in a published paper by improperly assuming the Limit Comparison Test to be valid for nonpositive series. Here are Dirichlet's two series: n=1 (-1)" √n n=1 (-1)" √n 1+ (-1)") √√n Explain how they provide a counterexample to the Limit Comparison Test when the series are not assumed to be positive.
44. Assumptions Matter In 1829, Lejeune Dirichlet pointed out that the great French mathematician Augustin Louis Cauchy made a mistake in a published paper by improperly assuming the Limit Comparison Test to be valid for nonpositive series. Here are Dirichlet's two series: n=1 (-1)" √n n=1 (-1)" √n 1+ (-1)") √√n Explain how they provide a counterexample to the Limit Comparison Test when the series are not assumed to be positive.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:44. Assumptions Matter In 1829, Lejeune Dirichlet pointed out that
the great French mathematician Augustin Louis Cauchy made a mistake
in a published paper by improperly assuming the Limit Comparison Test
to be valid for nonpositive series. Here are Dirichlet's two series:
IM8
Σ
(-1)"
√n
Σ
n=1
(-1)"
√n
(-1)"
√n
Explain how they provide a counterexample to the Limit Comparison
Test when the series are not assumed to be positive.
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