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forum.math.uoa.gr • Προβολή θέματος - Άπειρο

forum.math.uoa.gr

Forum του Τμήματος Μαθηματικών
Ημερομηνία 18 Νοέμ 2017, 06:15

Όλοι οι χρόνοι είναι UTC + 2 ώρες [ DST ]




Δημιουργία νέου θέματος Απάντηση στο θέμα  [ 2 δημοσιεύσεις ] 
Συγγραφέας Μήνυμα
 Θέμα δημοσίευσης: Άπειρο
ΔημοσίευσηΔημοσιεύτηκε: 11 Μάιος 2017, 00:27 
Χωρίς σύνδεση

Εγγραφή: 10 Μαρ 2016, 14:08
Δημοσ.: 33
Από διαγωνισμό :

Έστω συνεχώς παραγωγίσιμη συνάρτηση [unparseable or potentially dangerous latex formula] τέτοια, ώστε

[unparseable or potentially dangerous latex formula] και [unparseable or potentially dangerous latex formula] .

Να αποδειχθεί ότι [unparseable or potentially dangerous latex formula]


Κορυφή
 Προφίλ  
 
 Θέμα δημοσίευσης: Re: Άπειρο
ΔημοσίευσηΔημοσιεύτηκε: 12 Μάιος 2017, 00:32 
Χωρίς σύνδεση
Regular Forumer

Εγγραφή: 05 Φεβ 2008, 03:03
Δημοσ.: 419
Από την ανισότητα Cauchy-Schwartz, για κάθε [unparseable or potentially dangerous latex formula] με [unparseable or potentially dangerous latex formula], έχουμε ότι

[unparseable or potentially dangerous latex formula][unparseable or potentially dangerous latex formula][unparseable or potentially dangerous latex formula][unparseable or potentially dangerous latex formula],

αφού η f είναι γνησίως αύξουσα στο [unparseable or potentially dangerous latex formula], άρα [unparseable or potentially dangerous latex formula] για κάθε n>2. Eπομένως [unparseable or potentially dangerous latex formula].

Η τελευταία ανισότητα συνεπάγεται ότι [unparseable or potentially dangerous latex formula] για κάθε [unparseable or potentially dangerous latex formula] με [unparseable or potentially dangerous latex formula], επομένως, εφαρμόζοντάς τη για n=2^k, [unparseable or potentially dangerous latex formula], όπου k\in\mathbb N, έχουμε ότι [unparseable or potentially dangerous latex formula]. Επομένως, [unparseable or potentially dangerous latex formula].

Η τελευταία σειρά αποκλίνει όταν k\to\infty, άρα [unparseable or potentially dangerous latex formula].

_________________
\emptyset\not=\{\emptyset\}


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