※ 引述《KOREALee (韩国最高)》之铭言： : 标题: [微积] 证明数列递增且有界 : 时间: Mon May 30 19:13:49 2011 : : Let the sequence : : : 1 n : An = (1+ ---) : n : : : 1.show that An is increasing : : 2.show that An is bounded 1 1. By comparison with ── , we find when N >= No n^2 ∞ 1 Σ ─── is conv. n=N n! Since ∞ 1 Σ ─── is conv. (by p-series) n=N n^2 ∞ 1 2. Denote Σ ─── is A n=0 n! 1 Consider An=(1 + ──)^n n n n 1 n 1 n 1 = C + C *(──)^1 + ... + C *(──)^k + ... + C *(──)^n 0 1 n k n n n So we have n*(n-1)*...*(n-k+1) 1 = 1 + 1 + ... + ──────────*(──)^k + ... k! n where n*(n-1)*...*(n-k+1) 1 1 n n-1 n-k+1 ──────────*(──)^k = ── * (──) * (──) *...*(───) k! n k! n n n 1 1 k-1 = ── * 1 * (1- ──) *...*(1- ──) k! n n So we can rewrite An as： 1 1 1 1 2 An = 2 + ──*(1- ──) + ──*(1- ──)*(1- ──) +... 2! n 3! n n 1 1 k-1 + ── * (1- ──) *...*(1- ──) + ... k! n n 1 1 n-1 + ── * (1- ──) *...*(1- ──) n! n n 1 < e (Since each item is smaller then ──) k! So far we have showed that An is bounded and An+1 > An is trivial (by comparing each item) --

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