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Bài toán sẽ được chứng minh nếu ta có:
$\left( 1+\frac{1}{n}\right)^n<\left( 1+\frac{1}{n+1}\right)^{n+1}$
$\Leftrightarrow \frac{(n+1)^n}{n^n}<\frac{(n+2)^{n+1}}{(n+1)^{n+1}}$
$\Leftrightarrow (n+1)^{2n+1}<n^n(n+2)^{n+1}$
Ta có $(n+1)^2<n(n+1)$ nên $(n+1)^{2n+1}<n^n(n+2)^n(n+1)<n^n(n+2)^{n+1}$. (đpcm)
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