数列极限
$$\displaystyle若a_n>0(n = 1,2,\cdots),且\lim_{n\to\infty} \frac{a_n}{a_{n + 1}} = l > 1,则\lim_{n\to\infty}a_n = 0$$
$$\displaystyle若a_n>0(n = 1,2,\cdots),且\lim_{n\to\infty} \frac{a_{n + 1}}{a_{n}} = a,则\lim_{n\to\infty}\sqrt[n]{a_n} = 0$$
$$\displaystyle设\lim_{n\to\infty}(a_1 + a_2+\cdots + a_n)存在，证明：$$
$$\displaystyle(1)\lim_{n\to\infty}\frac{1}{n}(a_1+2a_2+\cdots+na_n) = 0$$
$$\displaystyle(2)\lim_{n\to\infty}(n!\cdot a_1a_2\cdots a_n)^\frac{1}{n} = 0(a_i > 0,i=1,2,\cdots,n)$$
$$\displaystyle已知\lim_{n\to\infty}a_n = a,\lim_{n\to\infty}b_n = b,证明：$$
$$\displaystyle \lim_{n\to\infty} \frac{a_1b_n +a_2b_{n-1}+\cdots + a_nb_1}{n} = ab$$
$$\displaystyle 设数列{a_n}满足\lim_{n\to\infty} \frac{a_1 +a_2+\cdots + a_n}{n}=a(-\infty<a<+\infty),证明\lim_{n\to\infty} \frac{a_n}{n}=0$$
$$\displaystyle 设0<\lambda<1,\lim_{n\to\infty}a_n = a,证明:$$
$$\displaystyle\lim_{n\to\infty}(a_n + \lambda a_{n-1} + \lambda^2 a_{n - 2} + \cdots + \lambda^n a_0) = \frac{a}{1 - \lambda}$$
$$\displaystyle A_n = \sum_{k = 1}^{n} a_n,当n\to\infty时有极限，{p_n}为严格单调递增的正数数列，且p_n\to +\infty(n\to\infty),证明：$$
$$\displaystyle \lim_{n\to\infty} \frac{p_1a_1+p_2a_2+\cdots+p_na_n}{p_n} = 0$$