n^k/a^n

Transcript

1. lim n→∞ nk an April 21, 2017
2. n, k : ࣗવ਺ a : ਖ਼ͷ࣮਺ ΋͠ a ≤ 1 ͳΒ໌Β͔ʹ nk/an → ∞ (n → ∞) ͳͷͰɺa > 1 ʹର ͯ͠ lim n→∞ nk an = 0 Λূ໌͠·͢ɻ
3. a > 1 ͳͷͰ a = 1 + h (h > 0) ͱ͓͚·͢ɻೋ߲ఆཧΑΓ an = (1 + h)n = nC01 + nC1h + nC2h2 + · · · + nCnhn Ͱ͕͢ɺn → ∞ ͱ͢ΔͷͰ k < n ͱԾఆ͍͍ͯ͠Ͱ͢ɻ͜ͷͱ͖ an = nC0 + · · · + nCk+1hk+1 + · · · + nCnhn ≥ nCk+1hk+1 ͕੒Γཱͪ·͢ɻ
4. an ≥ nCk+1hk+1 ͕෼͔ͬͨͷͰ 0 ≤ nk ak ≤ nk nCk+1hk+1 = nk n! (n − (k + 1))!(k + 1)! hk+1 = nk n(n − 1) · · · (n − k) (k + 1)! hk+1 = 1 n(1 − 1/n) · · · (1 − k/n) (k + 1)! hk → 0 (n → ∞) ͱͳͬͯɺ lim n→∞ nk an = 0 ͕ূ໌͞Ε·ͨ͠ɻ