Problem Sets: Sequences and Their Limits (数列及其极限)

Last Update: 2025-03-04

Remark:

There is a close connection between the limits of sequences and the limits of functions. If a sequence ${a_n}$ can be explicitly written in the form $a_n = f(n)$, determining its convergence or divergence and calculating its limit are often analogous to the corresponding procedures for the function $f(x)$. We may therefore use methods from function limits to analyze such sequences, which can be reviewed in the Calculus I material.

In these exercises, we avoid template‑based tools like the Stolz theorem, as they do not help develop a genuine understanding of limits, convergence, and approximation.

• A sequence ${a_n}$ is called a Cauchy sequence if for any $\varepsilon > 0$, there exists an $N \in \mathbb{N}^{+}$ such that\(\lvert a_n - a_m\rvert < \varepsilon \quad \text{for all } m, n > N.\)
  • Show that ${a_n}$ converges $\iff$ ${a_n}$ is a Cauchy sequence.

• Let $\displaystyle \lim_{n\to \infty} a_n = a$, prove that \(\displaystyle \lim_{n\to \infty}\frac{a_1 + a_2 + \cdots + a_n}{n} = a.\)

• Find $\displaystyle \lim_{n\rightarrow \infty} n\left(\frac{1}{n^{2}} + \dfrac{1}{n^{2}+1} + \cdots + \dfrac{1}{n^{2}+n}\right)$

• Let $\displaystyle x_n = (1 + a)(1 + a^{2})\cdots(1 + a^{2^{n}})$, where $\lvert a \rvert < 1$. Find $\displaystyle \lim_{n\to \infty} x_n$.

• Find $\displaystyle \lim_{n \to \infty} \sqrt[3]{n} \int_{n}^{1+n} \frac{\sin t}{\sqrt{t + \cos t}} \, dt$

• Let $\displaystyle x_1 = 2026, x_n^{2} - 2(x_n + 1)x_{n+1} + 2026 = 0$ ($n \ge 1$). Show that sequence ${x_n}$ converges, and find $\displaystyle \lim_{n \to \infty} x_n$.

• Find $\displaystyle \lim_{n \to \infty} (n!)^{\frac{1}{n^{2}}}$

• Find $\displaystyle \lim_{n \to \infty} \frac{1^k + 3^k + \cdots + (2n - 1)^k}{n^{k+1}} \quad (k > 0)$

• Find $\displaystyle \lim_{n \to \infty} \left[(n!)^{-1} \cdot n^{-n} \cdot (2n)!\right]^{\frac{1}{n}}$

• Let $f(x)$ have a continuous derivative on $[0,1]$, with $f(0)=0$ and $f(1)=1$. Prove that $\displaystyle \lim_{n \to \infty} n \left( \int_0^1 f(x) \, dx - \frac{1}{n} \sum_{k=1}^n f\left( \frac{k}{n} \right) \right) = -\frac{1}{2}$.

• Find $\displaystyle \lim_{n\to \infty}\left[\sqrt[n+1]{(n+1)!} - \sqrt[n]{n!}\right]$

• Find $\displaystyle \lim_{n \to \infty} n^2 \left[ \left(1 + \frac{1}{n}\right)^{n+1} - \left(1 + \frac{1}{n}\right)^n \right]$

• Consider the recursive sequence ${x_n}$ defined by \(x_{n+1} = f(x_n) \quad \text{for all } n \in \mathbb{N}^{+}.\)Prove that if the function $f$ is a contraction mapping—i.e., there exists a constant $L \in [0,1)$ such that \(\displaystyle \lvert f(x) - f(y)\rvert \leq L\lvert x - y\rvert \quad \text{for all } x, y \in \operatorname{dom}(f),\)then the sequence ${x_n}$ converges.

• Let $x_1 = 1$ and $\displaystyle x_{n+1} = 1 + \frac{1}{x_n}$ for $n \ge 1$. Prove that the limit of the sequence ${x_n}$ exists, and find $\displaystyle \lim_{n \to \infty} x_n$.