



 
Rahul, It is not a series, because a_n is not obtained by adding one more explicit term to $a_(n1).$ For instance: $a_1 = 1/(1+1) = 1/2$ $a_2 = 1/(1+2) + 1/(2+2) = 1/3 + 1/4$ $a_3 = 1/(1+3) + 1/(2+3) + 1/(3+3) = 1/4 + 1/5 + 1/6$ $a_4 = 1/(1+4) + 1/(2+4) + 1/(3+4) + 1/(4+4) = 1/5 + 1/6 + 1/7 + 1/8$ If it were a series then (eg) $a_4$ would be $a_3$ with exactly one new term added on and no other changes  but it isn't. To guess whether it is increasing or decreasing, evaluate these terms. To understand it, experiment: write out (say) $a_9$ and $a_{10}.$ Then find and simplify $a_{10}  a_9.$ To prove it, do the same thing using $a_n$ and $a_{n+1}.$ Good Hunting!  


Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences. 