Questions tagged [floor-function]

The floor function, also known as the greatest integer function, maps a real number $x$ to the greatest integer less than or equal to $x$ (often denoted $\lfloor x \rfloor$). See also (ceiling-function).

1,186 questions
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Are there ways to solve miscellaneous equations such as $\sin x=\log [x]$ without drawing the graphs?

Consider the example $$\sin x=\log [x]$$ where $[\,·\,]$ represents Greatest Integer Function. It is a miscellaneous equation, and I have been told that the only way to solve it is to draw the graphs ...
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Which of the following relations are true?Here $[x]$ denotes the greatest integer less then or equal to $x$ (as in option b) symbol

Which of the following relations are true? $1)$ $(-1)^{\frac{n(n-1)}{2}} = (-1)^{\frac{n(n+1)}{2}}$ $2)$ $(-1)^{\frac{n(n-1)}{2}} = (-1)^{[\frac{n}{2}]}$ $3)$ $(-1)^{\frac{n(n-1)}{2}} = (-1)^{n^2}$ ...
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The priority of limits

How are the two expressions different? $$\lim_{x\to0}\bigg\lfloor\frac{\sin{x}}{x}\bigg\rfloor$$ and $$\bigg\lfloor\lim_{x\to0}\frac{\sin{x}}{x}\bigg\rfloor$$ If limit is inside the floor function, ...
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Find the cardinality of the set $A_p$ defined as the following : [duplicate]

For any prime number $p$, $A_p$=the set of integers $d\in \{1,2,3,\dots, n\}$ such that the power of $p$ in the prime factorization of $d$ is odd. Then \begin{align*} A_p= & \lfloor\dfrac{n}{p}\...
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$\lim_{n \to \infty} \sum_{k=1}^{\left\lfloor\frac{n}{2}\right\rfloor}{\binom{n-k}{k}\frac{1}{2^{n-k}}}$?

Consider the following limit: $$\lim_{n \to \infty} \sum_{k=0}^{\left\lfloor\frac{n}{2}\right\rfloor}{\binom{n-k}{k}\frac{1}{2^{n-k}}}.$$ I can find the limit numerically, but is it possible to ...
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Find the number of possible solution of |[x]-2x|=4

Find the number of possible values of x of the equation |[x]-2x|=4. |x| represent absolute value of x. [x] represent greatest integer lesser than x. I plotted the curve in desmos.com and got the ...