Question

the nth term of the series 1²/3+ 1²+2²/5+ 1²+2²+3²/7+ ........... infinity ​

Answer 1

Answer:

HLet's write an expression for the nth term in the numerator 3,5,7……. It is an AP with the first term(a)=3,common difference (d)=2 so Tn=a+(n-1)d, Tn=3+(n-1)2= 2n+1.Now let’s write an expression for the nth term in the denoninator. In the first term it is 1^2, in the second term it is 1^2+2^2. So in the nth term it will be the sum of 1^2+2^2+3^3+……..n^2. As we know it will be n(n+1)(n+2)/ 6 .So by dividing the respective nth term of numerator and denominator we will get the answer. It is 6/n(n+1).