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]↺↻)

E71

[0CP]↺↻

E71

[0CR]↺↻

E71

[0CS]↺↻

E71

[0CV]↺↻

E71

[0CX]↺↻

E71

[0D0]↺↻

E71

[0D2]↺↻

E71

[0D4]↺↻

E71

[0D6]↺↻

E71

[0D9]↺↻

E71

[0DD]↺↻

See also exercises [0B9]↺↻ and [0B7]↺↻.

Summation by parts

E73

[217]↺↻

E73

[21H]↺↻

7.2 Recursive sequences

E73

[0DK]↺↻

E73

[0DN]↺↻

7.3 Series

Tests

The Dirichlet criteria implies the Liebniz “alternating series test” criteria.

In particular, if we set \(a_{n}=(-1)^{n}\) we prove the existence of the limit in Leibniz test.

Exercises

E80

[214]↺↻

E80

[23D]↺↻

E80

[0DW]↺↻

E80

[0DY]↺↻

E80

[0F0]↺↻

E80

[0F2]↺↻

E80

[0F4]↺↻

E80

[0F5]↺↻

E80

[0F7]↺↻

E80

[0F8]↺↻

E80

[0F9]↺↻

E80

[21M]↺↻

E80

[23F]↺↻

E80

[20Z]↺↻

E80

[210]↺↻

See also exercise [1T9]↺↻.

Cauchy product

E81

[0FJ]↺↻

E81

[0FK]↺↻

E81

[0FM]↺↻

E81

[0FP]↺↻

See also exercise [1KQ]↺↻.

7.4 Generalized sequences, or “nets’

[29X]↺↻

7.5 Generalized series

Generalized series with positive terms

E82

[0FX]↺↻

E82

[0FZ]↺↻

E82

[0G0]↺↻

E82

[0G3]↺↻

[ [0G4]↺↻]