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$$ \lim_{n\to \infty} \left(2+\frac{1}{n}\right)^{n} = ? $$

I don't know even how to start this.

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3 Answers 3

up vote 4 down vote accepted

Just look at it. It blows up.${}{}{}{}{}{}$

And fast! It's bigger than $2^n$.

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Since $2+1/n > 2 $ for all $n$, we have $(2+1/n)^n > 2^n$ for each $n$, so the sequence tends to infinity.

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$$\left(2+\frac{1}{n}\right)^n=\left(1+\left[1+\frac{1}{n}\right]\right)^n=\sum_{k=0}^n\binom{n}{k}\left[1+\frac{1}{n}\right]^k\geq\frac{n(n-1)}{2}\left(1+\frac{1}{n}\right)^2\xrightarrow [n\to\infty]{}\infty$$

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