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Chapter 9: Problem 19

Find the limit of the following sequences or determine that the limit does not exist. $$\left\\{\left(1+\frac{2}{n}\right)^{n}\right\\}$$

Short Answer

Expert verified
Answer: The limit of the sequence as n approaches infinity is e.

Step by step solution

01

Rewrite the sequence

Rewrite the sequence in a more familiar form: $$\left\\{\left(1+\frac{2}{n}\right)^{n}\right\\} = \left\\{\left(1+\frac{1}{n/2}\right)^{n}\right\\}$$
02

Apply the squeeze theorem

Observe that the given sequence is always positive since it is raised to the power of \(n\). We need to find two different sequences that converge to the same limit that will "squeeze" the given sequence. Recall that \(\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n} = e\). We will use this limit to find the two sequences. Our sequence lies between two sequences: $$\left\\{\left(1+\frac{1}{n}\right)^{n}\right\\} \leq \left\\{\left(1+\frac{1}{n/2}\right)^{n}\right\\} \leq \left\\{\left(1+\frac{1}{n/2}\right)^{2n}\right\\}$$ Now, to apply the squeeze theorem, we need to find the limit as \(n\) approaches infinity.
03

Find the limit of the bounding sequences

We already know the limit of the left sequence: $$\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n} = e$$ For the right sequence, the limit is: $$\lim_{n\to\infty}\left(1+\frac{1}{n/2}\right)^{2n} = \lim_{n\to\infty}\left[\left(1+\frac{1}{n/2}\right)^{n}\right]^2$$ Since \(\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n} = e\), we substitute \(\frac{1}{n/2}\) for \(\frac{1}{n}\): $$\lim_{n\to\infty}\left[\left(1+\frac{1}{n/2}\right)^{n}\right]^2 = e^2$$
04

Conclude the limit with the squeeze theorem

Using the squeeze theorem, we have: $$e \leq \lim_{n\to\infty}\left(1+\frac{2}{n}\right)^{n} \leq e^2$$ Since both bounds converge to the same limit (e), by the squeeze theorem, the limit of the given sequence is also e: $$\lim_{n\to\infty}\left(1+\frac{2}{n}\right)^{n} = e$$

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Most popular questions from this chapter

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