MathDB

2

Part of 2005 Grigore Moisil Urziceni

Problems(3)

Three nice limits

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11/28/2019
a) Prove that lim⁡x→∞x⋅∑k=1⌊x⌋1k+x=1. \lim_{x\to\infty } \sqrt{x}\cdot\sum_{k=1}^{\lfloor \sqrt{x} \rfloor} \frac{1}{k+x}=1.
b) Show that lim⁡x→∞(−⌊x⌋+x⋅∑k=1⌊x⌋1k+x)=−12 \lim_{x\to\infty } \left( -\left\lfloor\sqrt{x}\right\rfloor +x\cdot\sum_{k=1}^{\lfloor \sqrt{x} \rfloor} \frac{1}{k+x} \right) =\frac{-1}{2}
c) What about lim⁡x→∞(−x+x⋅∑k=1⌊x⌋1k+x)? \lim_{x\to\infty } \left( -\sqrt{x} +x\cdot\sum_{k=1}^{\lfloor \sqrt{x} \rfloor} \frac{1}{k+x} \right) ?
limitsreal analysis
Characterization of geometric progressions of length 3 between 2004² and 2005²

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11/28/2019
Find all triples (x,y,z) (x,y,z) of natural numbers that are in geometric progression and verify the inequalities 4016016≤x<y<z≤4020025. 4016016\le x<y<z\le 4020025.
inequalitiesgeometric sequencenumber theory
A classic result in antiderivative theory

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11/28/2019
Let be a function f:R⟶R≥0 f:\mathbb{R}\longrightarrow\mathbb{R}_{\ge 0} that admits primitives and such that lim⁡x→0f(x)x=0. \lim_{x\to 0 } \frac{f(x)}{x} =0. Prove that the function g:R⟶R, g:\mathbb{R}\longrightarrow\mathbb{R} , defined as g(x)={f(x)/x,emsp;x≠00,emsp;x=0, g(x)=\left\{ \begin{matrix} f(x)/x ,&&emsp; x\neq 0\\ 0,& &emsp; x=0 \end{matrix} \right. , is primitivable.
functioncalculusreal analysisprimitives