Comportement global d'une suite - Spécialité

Suite majorée/minorée/bornée

Exercice 1 : Démontrer la convergence d'une suite (Niv 2)

Soit \(\left(u_n\right)\) une suite minorée décroissante.
Démontrer que \(\left(u_n\right)\) converge.

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Exercice 2 : Démontrer la convergence d'une suite (Niv 3)

Soit \[\left(u_n\right) : u_n = 7 - \dfrac{5}{n} \]Calculer \(u_{n+1} - u_n\) en fonction de n.

Démontrer que \(\left(u_n\right)\) converge.
Si plusieurs blocs "On sait que, or, donc" sont nécessaires, il faut les écrire à la suite les uns des autres et non imbriqués les uns dans les autres.

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Exercice 3 : Démontrer la convergence d'une suite (Niv 1)

Soit \(\left(u_n\right)\) une suite majorée croissante.
Démontrer que \(\left(u_n\right)\) converge.

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Exercice 4 : Démontrer la convergence d'une suite (Niv 2)

Soit \(\left(u_n\right)\) une suite minorée décroissante.
Démontrer que \(\left(u_n\right)\) converge.

Exercice 5 : Démontrer la convergence d'une suite (Niv 3)

Soit \[\left(u_n\right) : u_n = 8 - \dfrac{7}{n} \]Calculer \(u_{n+1} - u_n\) en fonction de n.

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null}]}, "SequenceProperty_Minorated": {"compound": {"blocks": {"SequenceProperty": {"previous_statement": true, "next_statement": true, "colour": 210, "inputs": [{"type": "dummy_input", "fields": [{"text": "(u\u2099) est", "type": "text"}, {"name": "property_type", "values": [["convergente", "Converges"], ["croissante", "Increasing"], ["d\u00e9croissante", "Decreasing"], ["major\u00e9e", "Majorated"], ["minor\u00e9e", "Minorated"]], "type": "dropdown"}], "align": "left"}]}}, "ast": {"type": "SequenceProperty", "property_type": "Minorated"}}, "previous_statement": true, "next_statement": true, "colour": 210}, "SimpleComparison": {"previous_statement": true, "next_statement": true, "colour": 210, "inputs": [{"type": "value_input", "fields": [], "align": "left", "name": "value_1", "check": null}, {"type": "dummy_input", "fields": [{"name": "op", "values": [["=", "EQ"], ["\u2260", "NE"], ["<", "LT"], ["\u2264", "LE"], [">", "GT"], ["\u2265", "GE"], ["appartient \u00e0", "BELONGS"]], "type": "dropdown"}], "align": "left"}, {"type": "value_input", "fields": [], "align": "left", "name": "value_2", "check": null}]}, "SequenceProperty_Majorated": {"compound": {"blocks": {"SequenceProperty": {"previous_statement": true, "next_statement": true, "colour": 210, "inputs": [{"type": "dummy_input", "fields": [{"text": "(u\u2099) est", "type": "text"}, {"name": "property_type", "values": [["convergente", "Converges"], ["croissante", "Increasing"], ["d\u00e9croissante", "Decreasing"], ["major\u00e9e", "Majorated"], ["minor\u00e9e", "Minorated"]], "type": "dropdown"}], "align": "left"}]}}, "ast": {"type": "SequenceProperty", "property_type": "Majorated"}}, "previous_statement": true, "next_statement": true, "colour": 210}, "Start": {"next_statement": true, "colour": 65, "tooltip": "D\u00e9marrage de la d\u00e9monstration", "inputs": [{"type": "dummy_input", "fields": [{"text": "D\u00e9monstration", "type": "text"}], "align": "left"}], "creatable": false, "on_event": "start"}, "SequenceProperty_Converges": {"compound": {"blocks": {"SequenceProperty": {"previous_statement": true, "next_statement": true, "colour": 210, "inputs": [{"type": "dummy_input", "fields": [{"text": "(u\u2099) est", "type": "text"}, {"name": "property_type", "values": [["convergente", "Converges"], ["croissante", "Increasing"], ["d\u00e9croissante", "Decreasing"], ["major\u00e9e", "Majorated"], ["minor\u00e9e", "Minorated"]], "type": "dropdown"}], "align": "left"}]}}, "ast": {"type": "SequenceProperty", "property_type": "Converges"}}, "previous_statement": true, "next_statement": true, "colour": 210}, "theorem_GeIncreasing": {"previous_statement": true, "next_statement": true, "colour": 23, "inputs": [{"type": "dummy_input", "fields": [{"text": "Par d\u00e9finition d'une suite", "type": "text"}], "align": "left"}, {"type": "dummy_input", "fields": [{"text": "croissante", "type": "text"}], "align": "left"}]}, "theorem_MajoratedIncreasingConverges": {"previous_statement": true, "next_statement": true, "colour": 23, "inputs": [{"type": "dummy_input", "fields": [{"text": "Une suite major\u00e9e croissante", "type": "text"}], "align": "left"}, {"type": "dummy_input", "fields": [{"text": "est convergente", "type": "text"}], "align": "left"}]}, "theorem_LeMajorated": {"previous_statement": true, "next_statement": true, "colour": 23, "inputs": [{"type": "dummy_input", "fields": [{"text": "Par d\u00e9finition d'une suite", "type": "text"}], "align": "left"}, {"type": "dummy_input", "fields": [{"text": "major\u00e9e", "type": "text"}], "align": "left"}]}, "c": {"compound": {"blocks": {"base_stringinput": {"output": "String", "colour": 260, "tooltip": "Fourni une chaine de caract\u00e8re", "inputs": [{"type": "dummy_input", "fields": [{"name": "value", "default": "", "type": "text_input"}], "align": "left"}]}}, "ast": {"type": "base_stringinput", "value": "u\u2099"}}, "output": "String", "colour": 260, "tooltip": "Fourni une chaine de caract\u00e8re"}, "B": {"compound": {"blocks": {"base_stringinput": {"output": "String", "colour": 260, "tooltip": "Fourni une chaine de caract\u00e8re", "inputs": [{"type": "dummy_input", "fields": [{"name": "value", "default": "", "type": "text_input"}], "align": "left"}]}}, "ast": {"type": "base_stringinput", "value": "0"}}, "output": "String", "colour": 260, "tooltip": "Fourni une chaine de caract\u00e8re"}, "SequenceProperty_Increasing": {"compound": {"blocks": {"SequenceProperty": {"previous_statement": true, "next_statement": true, "colour": 210, "inputs": [{"type": "dummy_input", "fields": [{"text": "(u\u2099) est", "type": "text"}, {"name": "property_type", "values": [["convergente", "Converges"], ["croissante", "Increasing"], ["d\u00e9croissante", "Decreasing"], ["major\u00e9e", "Majorated"], ["minor\u00e9e", "Minorated"]], "type": "dropdown"}], "align": "left"}]}}, "ast": {"type": "SequenceProperty", "property_type": "Increasing"}}, "previous_statement": true, "next_statement": true, "colour": 210}, "theorem_MinoratedDecreasingConverges": {"previous_statement": true, "next_statement": true, "colour": 23, "inputs": [{"type": "dummy_input", "fields": [{"text": "Une suite minor\u00e9e d\u00e9croissante", "type": "text"}], "align": "left"}, {"type": "dummy_input", "fields": [{"text": "est convergente", "type": "text"}], "align": "left"}]}, "theorem_GeMinorated": {"previous_statement": true, "next_statement": true, "colour": 23, "inputs": [{"type": "dummy_input", "fields": [{"text": "Par d\u00e9finition d'une suite", "type": "text"}], "align": "left"}, {"type": "dummy_input", "fields": [{"text": "minor\u00e9e", "type": "text"}], "align": "left"}]}}, "toolbox": [{"type": "And"}, {"type": "Implication"}, {"type": "SequenceProperty_Converges"}, {"type": "SequenceProperty_Decreasing"}, {"type": "SequenceProperty_Increasing"}, {"type": "SequenceProperty_Majorated"}, {"type": "SequenceProperty_Minorated"}, {"type": "SimpleComparison", "value_1": {"type": "c"}, "value_2": {"type": "A"}, "op": "GE"}, {"type": "SimpleComparison", "value_1": {"type": "c"}, "value_2": {"type": "A"}, "op": "LE"}, {"type": "SimpleComparison", "value_1": {"type": "d"}, "value_2": {"type": "B"}, "op": "GE"}, {"type": "SimpleComparison", "value_1": {"type": "d"}, "value_2": {"type": "B"}, "op": "LE"}, {"type": "theorem_GeIncreasing"}, {"type": "theorem_GeMinorated"}, {"type": "theorem_LeDecreasing"}, {"type": "theorem_LeMajorated"}, {"type": "theorem_MajoratedIncreasingConverges"}, {"type": "theorem_MinoratedDecreasingConverges"}]}, "ast": [[{"type": "Start", "deletable": false, "movable": false, "editable": false}]]}

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