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On en déduit que lim sup u M. Soit ε ∈ ]0, M [. L’application f étant continue, par définition de M , il existe des réels c, d vérifiant a c < d b et f (t) M − ε pour tout t ∈ [c, d]. Alors d 1/n [f (t)]n dt un (d − c)1/n (M − ε). c D’où lim inf u M − ε. 6. M. lim sup u, il vient lim inf u = lim sup u, et on conclut d’après √ √ 3)n =√ N + N 3, avec N,√N ∈ N. On a alors (2 − 3)n = N − N √3. D’où π(2 + 3)n = 2πN − π(2 − 3)n . Par√suite, si l’on n pose vn = sin[π(2 √ − n 3) ], il vient un = −vn . Or, comme 0 < 2 − 3 < 1, on a 0 < vn < π(2 − 3) .

Si a > 1, dès que Rn−1 < 1, on a vn > un · Rn−1 D’où à nouveau divergence de la série d’après ce qui précède. 3. Supposons 0 < a < 1. On a : vn = On en déduit que, pour N Rn−1 − Rn α Rn−1 Rn−1 Rn dt . tα 1, R0 N vn n=1 d’après l’hypothèse sur α. La série 0 dt < +∞ tα vn est convergente. © Dunod. La photocopie non autorisée est un délit. 5. On a ou αn = αn+1 , ou αn+1 = 1 + αn . • Si αn = αn+1 , on a un+1 /un = a. Si αn+1 = 1 + αn , il vient un+1 /un = bαn +1 , et ce dernier terme peut être rendu arbitrairement grand avec n.

On a donc obtenu : Théorème. L’application (R, +) → (U, ×), t → eit est un homomorphisme de groupes, surjectif et non injectif. Son noyau est de la forme aZ, avec a ∈ R ∗+ . Le réel a qui est le plus petit réel positif t tel que e it = 1, est noté 2π . 6. (i) Le noyau de l’homomorphisme (C, +) → (C ∗ , ×) est 2iπZ. (ii) La fonction exponentielle complexe est périodique, et l’ensemble de ses périodes est 2πiZ. Démonstration. 5). La réciproque est analogue. (ii) Si z, ζ ∈ C, on a ez+ζ = ez si et seulement si eζ = 1.

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