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Three nice limits

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November 28, 2019
limitsreal analysis

Problem Statement

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) ?
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