Homework 2: Limits 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.


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