jasperE3 wrote: For each integer $n\ge2$ prove the inequality $$\log_23+\log_34+\ldots+\log_n(n+1)<n+\ln n-0.9.$$ Let n be a natural number and $ n\geq5$. Prove the inequality: $$ \log_23+\log_34+\cdots+\log_n(n+1)>n$$Study the convergence of the following sequence : \[ a_n=\log_2 3+\log_3 4+...+\log_n(n+1)-(n+\ln \ln(n))\]