收敛和发散的示例

For an example of a convergent sequence, let us examine $a_n = \left(1 + \dfrac{1}{n}\right)^n$, the well known sequence that converges to $e$, Euler's number.

This can be seen by rewriting the limit and using l'Hospital's Rule: $$\begin{align*} \lim_{n \to \infty} \left(1+\dfrac{1}{n}\right)^n &= \lim_{n \to \infty} e^{n\ln\left(1+\frac{1}{n}\right)} \\ &= e^{\lim_{n \to \infty} n\ln\left(1+\frac{1}{n}\right)} \\ &= e^{\lim_{n \to \infty} \dfrac{\ln\left(1+\frac{1}{n}\right)}{\frac{1}{n}}} \\ &= e^{\lim_{n \to \infty} \dfrac{\frac{1}{1+\frac{1}{n}}\cdot\left(-\frac{1}{n^2}\right)}{-\frac{1}{n^2}}} \\ &= e^{\lim_{n \to \infty} \dfrac{1}{1+\frac{1}{n}}} \\ &= e \end{align*}$$

$a_n = \dfrac{3n^4 + 34n^3 + 14}{2n^2 + 15n + 8}$ is a divergent sequence. This is clear because the expression is "top-heavy" because the degree of the numerator is greater than that of the denominator. To verify this, you can use l'Hospital's Rule repeatedly on the expression to determine the limit.

$\lim_{n \to \infty} \dfrac{A(n)}{B(n)} = \infty$ if the polynomial $A(x)$ is of a greater degree than $B(x)$. If we use l'Hospital's Rule repeatedly, the denominator eventually becomes a constant, while the numerator still contains some power of the variable.

Similarly, $\lim_{n \to \infty} \dfrac{A(n)}{B(n)} = 0$ if the degree of $A(x)$ is smaller than that of $B(x)$.

If the degrees are the same, the limit can be found by taking the ratio of the coefficients of the highest power of $n$. For example, $\lim_{n \to \infty} \dfrac{14x^3 + 8x^2 +67}{4x^3 + 45x + 37} = \dfrac{14}{4} = \dfrac{7}{2}$