Putnam preparation (Table of contents)

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.