Putnam preparation (Table of contents)

Real Analysis

Problem 1

Let \(f: [a,b]\to\mathbb R\) be a continuous function and \((a_n)_{n=1}^{\infty}\) a sequence of numbers from the interval \([a,b]\). Prove that there exists \(x\in[a,b]\) such that \[f(x)=\sum_{n=1}^{\infty} \frac{f(a_n)}{2^n}.\]

Problem 2

If \(f:[a,b]\to\mathbb R\) is a continuous function that is differentiable on \((a,b)\) prove that there exists \(c\in (0,1)\) such that \[f(1)-f(0)=\frac{f^{\prime}(c)+f^{\prime}(1-c)}2.\]

Problem 3

Find the limit \[\lim_{x\to 0}\frac{1-(\cos x)^{\sin x}}{x^3}.\]

Problem 4

  • (a) Assume that \((a_n)_{n=0}^{\infty}\) and \((b_n)_{n=0}^{\infty}\) are two sequences of integers. Prove the integration by parts formula for sequences: \[\sum_{n=1}^k a_n(b_{n}-b_{n-1})=a_kb_k-a_0b_0-\sum_{n=0}^{k-1}b_n(a_{n+1}-a_n).\]

  • (b) Assume that \((a_n)_{n=1}^{\infty}\) and \((b_n)_{n=1}^{\infty}\) are two sequences such that \((a_n)_{n=1}^{\infty}\) is decreasing, positive, and with limit \(0\), and \((b_n)_{n=1}^{\infty}\) is a sequence for which there exists \(M > 0\) such that \(\left|b_1+b_2+\cdots+ b_k\right| < M\) for all \(k\in\mathbb N\). Prove that the series \(\displaystyle\sum_{n=1}^{\infty}a_nb_n\) is convergent.

  • (c) Prove that the series \(\displaystyle\sum_{n=1}^{\infty}\frac{\sin n}{\sqrt n}\) is convergent.

Problem 5

Find the limit \[\lim_{n\to+\infty}\left(\frac1{n+1}+\frac1{n+2}+\cdots+\frac1{5n}\right).\]

Problem 6

Find the maximum value of \[\int_0^1 \left(x^2f(x)-xf^2(x)\right)\,dx\] where the maximum is taken over all continuous functions \(f:[0,1]\to\mathbb R\).

Problem 7

Assume that \(f:\mathbb R\to\mathbb R\) is a convex function that satisfies \(\displaystyle\lim_{x\to+\infty}\frac{f(x)}{x}=+\infty\). Prove that \[\lim_{x\to+\infty}\left(f(2x)-2f(x)\right)=+\infty.\]

Problem 8

Assume that \(f: \mathbb R_+\to\mathbb R_+\) is a differentiable function such that for all \(x > 0\) the following inequality is satisfied: \[xf^{\prime\prime}(x)+f^{\prime}(x)+f(x)\leq 0.\] Prove that \(\displaystyle \lim_{x\to+\infty}f(x)=0\).