Prove that \left( 1 + \frac{1}{n} \right)^n converges.
Solution
Let G(n) = \left( 1 + \frac{1}{n} \right)^n .
Binomial expansion, G(n) = \sum_{k = 0}^{n} \frac{ \binom{n}{k} }{n^k} .
By simple inequality argument, \frac{ \binom{n}{k} }{n^k} < \frac{1}{k!} .
So if we let F(n) = \sum_{k = 0}^{n} \frac{1}{k!} , we have G(n) < F(n) .
By simple inequality argument, F(n) < 1 + \sum_{k = 1}^{n} \frac{1}{2^{k-1}} < 3 .
So G and F are bounded above.
By AM-GM, \frac{G(n-1)}{G(n)} = \left( \frac{ n^2 }{ n^2 - 1} \right)^{(n-1)} \frac{ n}{n+1} < \left( \frac{ \frac{ n^2 }{ n^2 - 1} (n-1) + \frac{ n}{n+1} }{n} \right)^n = 1 . So G(n-1) < G(n) .
Since G is monotonic increasing and bounded above, G converges.
Similarly F converges.
Let's also prove that G and F converges to the same limit.
By studying individual terms in F(n) - G(n) and making simple inequality arguments, we can show that for each \epsilon > 0 we can pick N s.t. F(n) - G(n) < \epsilon for all n > N. This can be further used to make the formal argument that the limit of F is the same as the limit of G.
So if we call the limit e, then \lim_{n \to \infty } \left( 1 + \frac{1}{n} \right) ^n = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \cdots = e .
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