Question

I have a series An= (alpha*n)/(beta*n+1) and alpha is a real number, beta greater than 0, n greater or equal to 1 and i have to prove that it is bounded and monotonous

Answer 1

\[\alpha<0\]\[\alpha n +\alpha \beta n^2 + \alpha + \alpha \beta n < \alpha n +\alpha \beta n^2 + \alpha \beta n \]\[(\alpha n + \alpha)(1+\beta n)<(\alpha n)(1+ \beta n + \beta)\]\[\frac{ \alpha n + \alpha }{ 1+ \beta n + \beta }<\frac{ \alpha n }{ 1+ \beta n }\]\[\frac{ \alpha(n+1) }{ 1+\beta (n+1) }<\frac{ \alpha n }{ 1+ \beta n }\]\[A_{n+1}<A_n\] therefore An is monotonous. a similar result can be arrived when you change "<" to ">"

Answer 2

\[\large \left| \frac{ \alpha n }{ 1+\beta n } \right|<\left| \frac{ \alpha n }{ \beta n } \right|<\left| \frac{ \alpha }{ \beta } \right|\] so it is bounded.

Answer 3

Thank you very much