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 Dec 13 comment Prove that $\log{\frac{\sum_{j=1}^n x_j}{n}}\ge \frac{\sum_{j=1}^n \log x_j}n$ for $j=1,…,n$ Your ineq is just Jensen ineq applied to the concave function log(x) (I supposed log means ln or any logarithm with base greater than one) Dec 31 awarded Nice Answer Dec 14 comment Convergence of series involving iterated $\sin$ See also here (num.U213) awesomemath.org/wp-content/uploads/reflections/2012_1/… for a proof of the first two terms of the expansion Oct 15 awarded Yearling Nov 2 awarded Necromancer Oct 15 awarded Yearling Sep 7 comment How may I prove this inequality? Someone calls this method "uvw" downloadable here artofproblemsolving.com/Forum/… Sep 7 answered Compute $\int_{0}^{1}\frac{\ln(x) \ln^2 (1-x)}{x} dx$ Sep 2 comment proving :$\frac{(ab+b)(2b+1)}{(ab+a)(5b+1)}+\frac{(bc+c)(2c+1)}{(bc+b)(5c+1)}+\frac{(ca+a)(2a+1)}{(ca+c)(5a+1)}\ge\frac{3}{2}$ It comes from here fen.bilkent.edu.tr/~cvmath/Problem/problem_2011.htm (April 2011) Aug 30 answered Inequality $(a+\frac{1}{b})^2+(b+\frac{1}{c})^2+(c+\frac{1}{a})^2\ge 16$ Jun 21 answered Linear change of variables in Hamiltonian functions Jun 19 answered Question on sequences Jun 19 answered infinite series involving harmonic numbers and zeta Jun 14 answered Calculate alpha from $\alpha + \sin(\alpha)$ = K Jun 14 comment An inequality involving integrals @Mohamed: <<....function $x↦6x−2=3(2x−1)$ used...>>?? Jun 14 awarded Supporter Jun 14 answered An inequality involving integrals Jun 14 comment Convergence of a function series The general term may well be not positive: put a minus sign in front of the series. As for the monotonicity, note that $$\left(\frac 1t\ln \frac xt\right)' =\frac{\ln t}{t^2} -\frac{1+\ln x}{t^2}$$ so is asymptotically incresing. The assumption are satisfied except at most for a finite number of terms but this doesn't affect the convergence Jun 13 answered Convergence of a function series Jun 12 answered Computing $\sum_{n=0}^{\infty} u_{n}, \frac{u_{n+1}}{u_n}=\frac{n+a}{n+b}$