# Science:Math Exam Resources/Courses/MATH103/April 2014/Question 11

MATH103 April 2014
Other MATH103 Exams

### Question 11

'This is a difficult question and will be marked very strictly. You should try this ONLY AFTER you finish all other problems.

Suppose a sequence ${\displaystyle (b_{n})_{n\geq 0}}$ satisfies the recursive relation

${\displaystyle b_{n+1}=\beta {\frac {b_{n}}{1+b_{n}}},}$

where ${\displaystyle \beta }$ is a given positive constant. Suppose ${\displaystyle b_{0}=1}$. Find all positive constants ${\displaystyle \beta }$ for which the series ${\displaystyle \sum _{n=0}^{\infty }b_{n}}$ converges.