Introduction to e：練習問題

Evaluate the limit \( \lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^{2n} \).

Evaluate the limit:
$$\lim_{x\to 0} (1+3x)^{\frac{2}{x}}$$

Evaluate the limit:
\(\lim_{x \to 0} \left( \frac{e^{2x} + e^{4x}}{2} \right)^{\frac{1}{x}}\)

Simplify the expression:
$$\ln(e^5) + \ln\left(\frac{1}{e^2}\right)$$

Find the coefficient of \(x^2\) in the power series expansion of \(e^{3x} - e^{x}\).

Evaluate the limit:

Using the definition of the mathematical constant \(e\) as an infinite series, find the value of the constant \(k\) such that \(e^2 = \sum_{n=0}^{\infty} \frac{k^n}{n!}\).

Find the exact value of the sum \( \sum_{n=0}^{\infty} \frac{n+2}{n!} \) in terms of \( e \).

Consider the irrational number \( e \), which can be defined by the infinite series \( e = \sum_{k=0}^{\infty} \frac{1}{k!} \).

(a) Given the series expansion \( e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots \), show that for any positive integer \( n \), \( e > \left(1 + \frac{1}{n}\right)^n \) by considering the binomial expansion of \( \left(1 + \frac{1}{n}\right)^n \).
(b) Let \( a_n = \left(1 + \frac{1}{n}\right)^n \) and \( b_n = \left(1 + \frac{1}{n}\right)^{n+1} \). Show that \( a_n < e < b_n \) for all positive integers \( n \).
(c) Using the definition of \( e \) as a limit, evaluate \( \lim_{n \to \infty} \left( \frac{(n+1)^{n+1}}{n^n} - \frac{n^n}{(n-1)^{n-1}} \right) \) in terms of \( e \).

Let the power series expansion of \( e^x \) be given by \( e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \) for all real values of \( x \).

(a) Using differentiation, show that \( \sum_{n=1}^{\infty} \frac{n x^n}{n!} = x e^x \).

(b) Using the result in (a), or otherwise, show that \( \sum_{n=1}^{\infty} \frac{n^2 x^n}{n!} = (x^2 + x) e^x \).

(c) Hence, find the exact value of the infinite sum \( \sum_{n=1}^{\infty} \frac{n^2 - n + 1}{n!} \) in terms of \( e \).