# Convergence of Sequences and Series

Problem 1 The sequence $$(x_n)_{n=1}^{\infty}$$ is defined in the following way: $$x_1=1$$, and $$\displaystyle x_{n+1}= x_n\cdot \left(1-\frac1{n+1}\right)$$ for $$n\geq 2$$. Find the limit $$\displaystyle\lim_{n\to\infty} x_n$$ or prove that the limit does not exist.

Problem 2 Prove that the series $$\displaystyle\sum_{n=2}^{\infty}\frac2{(n-1)n(n+1)}$$ is convergent and find its value.

Problem 3

For each of the following statements determine whether it is true or false. If it is true, then prove it. If it is false, provide a counter-example.

• (a) If the series $$\displaystyle\sum_{n=1}^{\infty} a_n$$ converges, then $$\displaystyle\sum_{n=1}^{\infty} (-1)^na_n$$ converges.
• (b) If the series $$\displaystyle\sum_{n=1}^{\infty} a_n^2$$ converges, then $$\displaystyle\sum_{n=1}^{\infty} (-1)^n |a_n|^{\frac12}$$ converges.
• (c) If the series $$\displaystyle\sum_{n=1}^{\infty} a_n^2$$ converges, then $$\displaystyle\sum_{n=1}^{\infty} a_n^3$$ converges.

Problem 4

• (a) (Cesaro-Stolz theorem) If $$a_n$$ and $$b_n$$ are two sequences of real numbers such that $$b_n$$ is positive, increasing, and unbounded, prove that $\underline{\lim} \frac{a_{n+1}-a_n}{b_{n+1}-b_n}\leq \underline{\lim} \frac{a_n}{b_n}\leq \overline{\lim}\frac{a_n}{b_n}\leq \overline{\lim}\frac{a_{n+1}-a_n}{b_{n+1}-b_n}.$

• (b) Let $$x_n$$ be a sequence of real numbers such that $$\displaystyle\lim_{n\rightarrow\infty} (2x_{n+1}-x_n)=x$$. Show that $$\displaystyle\lim_{n\rightarrow\infty}x_n=x$$.

Problem 5

Let $$\varphi_0(x)=\ln x$$, and let $$\varphi_n(x)=\int_0^x \varphi_{n-1}(t)\,dt$$ for $$n\geq t$$.

• (a) Find the closed formula for $$\varphi_n(x)$$.

• (b) Find the limit $$\displaystyle \lim_{n\to \infty}\frac{n!\cdot \varphi_n(1)}{\ln n}$$.

Problem 6

Determine whether the series $\sum_{n=1000}^{\infty}\frac1{(\ln n)^{\ln(\ln n)} }$ is convergent or divergent.

Problem 7

Assume that $$p\in\mathbb N$$ and $$\varepsilon > 0$$. Prove that there exists positive integers $$m$$ and $$n$$ such that $\varepsilon < \left|\sqrt m-pn\right| < 2\varepsilon.$

Problem 8

Define the sequence $$(a_n)_{n=1}^{\infty}$$ in the following way: $a_n=\frac1{n\cdot \ln n\cdot \ln(\ln n)\cdot \ln(\ln(\ln n))\cdot \cdots \cdot \underbrace{\ln(\ln(\ln\cdots (\ln}_{k_n}(n))\cdots)},$ where $$k_n$$ is the largest integer such that $$\underbrace{\ln(\ln(\ln\cdots (\ln}_{k_n}(n))\cdots) > 1$$. Determine whether the series $$\displaystyle\sum_{n=3}^{\infty}a_n$$ is convergent.