Category: Relation and Functions

Question Number 73052 by mathmax by abdo last updated on 05/Nov/19 $$let{P}_n=X^n+X^{n-1}+\dots .+X^2+X-1\in R\left[X\right]$$$$1)provethat{P}_{n}haveoneroot{x}_{n}inside]0,+\mathrm{\infty }[$$$$2)studythesequence{x}_n$$ Answered…

Question Number 73046 by mathmax by abdo last updated on 05/Nov/19 $$provethat\mathrm{\forall }n\in N\sum _{k=0}^{n}k{C}_{2n}^{n+k}={nC}_{2n-1}^n$$ Answered by mind is power last…

Question Number 73047 by mathmax by abdo last updated on 05/Nov/19 $$solveinside{Z}^3{x}^2+y^2+z^2=2xyz$$ Answered by mind is power last updated…

Question Number 73044 by mathmax by abdo last updated on 05/Nov/19 $$provethat\sum _{k=1}^n{H}_k=(n+1)H_n-n$$$${and}\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{H}_{{k}} ^{\mathrm{2}} \:=\left({n}+\mathrm{1}\right){H}_{{n}} ^{\mathrm{2}} \:−\left(\mathrm{2}{n}+\mathrm{1}\right){H}_{{n}} \:+\mathrm{2}{n}…

Question Number 72990 by mathmax by abdo last updated on 05/Nov/19 $$calculate{lim}_{x\to 0}\frac{arctan\left(e^x\right)-\frac{\pi }{4}}{x^2}$$ Commented by abdomathmax last updated on 19/Nov/19 $${let}\:{use}\:{hospital}\:{theorem}\:\:{f}\left({x}\right)={arctan}\left({e}^{{x}}…