graydarl Group Title 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 one year ago one year ago

1. sirm3d Group Title

$\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 ">"

2. sirm3d Group Title

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

3. graydarl Group Title

Thank you very much