A sequence $u_1,u_2,u_3,\ldots$ is such that $u_1=1$ and \(u_{n+1}=u_n-\frac{2n+1}{n^2(n+1)

Question

A sequence $u_1,u_2,u_3,\ldots$ is such that $u_1=1$ and $u_{n+1}=u_n-\frac{2n+1}{n^2(n+1)^2}$, for all $n\geq1$. Use the method of mathematical induction to prove that $u_n=\frac1{n^2}$.[4] Hence find $\sum_{n=1}^N\frac{2n+1}{n^2(n+1)^2}$.[2] Give a reason why the series in part (ii) is convergent and state the sum to infinity. [2] Use your answer to part (ii) to find $\sum_{n=2}^N\frac{2n-1}{n^2(n-1)^2}$.[2]

Accepted Answer

(i) $u_n=1/n^2$. (ii) $1-1/(N+1)^2$. (iii) 1. (iv) $1-1/N^2$.

N2007 P2 Q2

N2007 P2 Q2

Solution:

Solution locked