Question

Hello Mathematicians,

Recently I have been doing some work which has an interesting application to another problem as a consequence. However many searches for this latter problem have not returned a proper name for it, so I would be very grateful if you could help...

Problem: Specify an infinite sequence $a_n(\lambda)$, dependent on both $n$ and $\lambda$ in a non-trivial way, such that

\begin{equation} \sum_n a_n(\lambda)^2=\rho\left(\sum_n a_n(\lambda)\right)^2, \end{equation}

where $\lambda$ may be altered to satisfy any finite value of the constant $\rho$.

I can solve this implicitly for $\rho\in\mathbb{R}$ under certain assumptions on the behaviour of $a_n(\lambda)$ and additionally can give methods for approximating $a_n(\lambda)$ to relatively high accuracy. The proof is a bit specialised and so I am mainly interested in more general work by others

Thank you for any help

Answer 1

If $a_n=a_n(\lambda)=\lambda^n$ then $\sum a_n^2=\frac{1}{1-\lambda^2}$ while $(\sum a_n)^2=\frac{1}{(1-\lambda)^2}$ So $\lambda=\frac{1-\rho}{1+\rho}$ makes $ \sum_n a_n(\lambda)^2=\rho\left(\sum_n a_n(\lambda)\right)^2$ for any $\rho \gt 0.$