7 Sequences and series[0CN]
7.1 Sequences
Let \((a_ n)_{n∈ℕ}⊆ ℝ\) be a real-valued sequence (as defined in [16G]).
Given \(N∈ ℕ\) we will write \(\sup _{n≥ N} a_ n\) in the following, instead of \(\sup \{ a_ N,a_{N+1}\ldots \} \), and similarly for the infimum. (This is in accordance with [20H])
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Exercise
72
Exercise
73
QuasiEsercizio
21
Summation by parts
QuasiEsercizio
22
7.2 Recursive sequences
7.3 Series
Tests
Theorem
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Root test
Theorem
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Ratio test
Remark
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Theorem
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Theorem
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Dirichlet criterion
Theorem
79
Alternating series test, or Leibniz test
Theorem
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Exercises
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QuasiEsercizio
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QuasiEsercizio
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QuasiEsercizio
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QuasiEsercizio
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QuasiEsercizio
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QuasiEsercizio
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QuasiEsercizio
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Cauchy product
Definition
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7.4 Generalized sequences, or “nets’
7.5 Generalized series
Generalized series with positive terms
Definition
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