Homework 4: 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 \( \displaystyle \lim_{x\to+\infty}f(x)=0 \).


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