Week 7 Exercises, due 9 November 2021

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Exercises 6.5 Week 7 Exercises, due 9 November 2021

For any \(n\in\NN\text{,}\) let \(a_n = \left(1 + \frac{1}{n}\right)^n\text{.}\) We have already seen that \(\sum_{j=0}^{\infty} 1/j!\) converges, and we called the limit \(e\) (or Euler's constant). Here, you will show that the sequence \((a_n)\) also converges, and the limit is again \(e\text{.}\)

1.

Use the AM/GM inequality to show that for every \(n\in\NN\text{,}\) one has \(a_n \lt a_{n+1}\text{.}\)

Hint.

Let \(x_1 = x_2 = \cdots = x_n = 1 + 1/n\) and \(x_{n+1}=1\text{.}\)

2.

Use the AM/GM inequality again to show that \(a_{2n+1} \lt 4\text{.}\) Deduce that the sequence \((a_n)\) converges.

Hint.

Let \(x_1 = x_2 = \cdots = x_n = 1\) and \(x_{n+1}=1/2\text{.}\)

3.

For every \(n\in\NN\text{,}\) prove that \(a_n \leq \sum_{j=0}^{n} 1/j!\text{.}\)

4.

Prove that \(\lim_{n \to \infty} a_n = \sum_{j=0}^{\infty} 1/j!\text{.}\)