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I came across the following problems during the course of my studying of real analysis:

Show that the sequence $(a_n)$ defined by $a_n = \left(1+ \frac{1}{n} \right)^{n}$ is bounded above by $3$.

I think we can use the binomial theorem. So $a_n = \left(1+ \frac{1}{n} \right)^{n} = \sum_{k=0}^{n} \binom{n}{k} \left(\frac{1}{n} \right)^{k}$

$= 1+ \sum_{k=1}^{n} \binom{n}{k} \left(\frac{1}{n} \right)^{k}$

From here, how would I deduce that this is $\leq 3$?

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    It would have been nice to see somewhat more computation. You mentioned an idea that you thought might be helpful. Indeed it is. But your questions are posted at such a fast rate that I think you may not be giving yourself enough time to seriously tackle each problem before seeking help.2011-06-22

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Just expand out the binomial coefficient as $\frac{(n)(n-1)\cdots (n-k+1)}{k!}$

Then you can conclude quickly that the sum is no greater than $1+\frac{1}{1!}+\frac{1}{2!}+ \cdots$

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    @Damien: That's exactly what my informal "Third term is $1/2$, fourth is <(1/2)^2, fourth is <(1/2)^3 and so on says".2011-06-23