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Chapter 9: Problem 20

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 given sequence as $n$ approaches infinity is $e^2$.

Step by step solution

01

Recognize the similarity with the Euler's number definition

The given sequence is similar to the definition of Euler's number, but with a slight difference. We can rewrite the given sequence as: $$\lim_{n\to\infty}\left(1+\frac{1}{\frac{n}{2}}\right)^{n}$$
02

Use substitution for simplicity

Let's make a substitution, let \(m = \frac{n}{2}\). Then as \(n \to \infty\), we have \(m \to \infty\). So our limit becomes: $$\lim_{m\to\infty}\left(1+\frac{1}{m}\right)^{2m}$$
03

Apply the Euler's number definition

Now, we can apply the definition of Euler's number: $$\lim_{m\to\infty}\left[\left(1+\frac{1}{m}\right)^{m}\right]^2$$ We know that as \(m \to \infty\), \(\left(1+\frac{1}{m}\right)^{m}\) tends to \(e\). Therefore, our limit becomes: $$\lim_{m\to\infty} e^2$$
04

Evaluate the limit

Since our limit does not depend on \(m\) anymore, it's just a constant value, which is simply: $$e^2$$ So, the limit of the given sequence is \(e^2\).

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