Creating an Infinite Series with a Specific Property [closed]

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

$$\sum_n a_n(\lambda)^2=\rho\left(\sum_n a_n(\lambda)\right)^2,$$

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

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closed as not a real question by Igor Rivin, Yemon Choi, David Loeffler, Simon Thomas, Gerry MyersonJun 1 '11 at 23:55

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

What are $\lambda$ and $\rho$ in this question? Parameters? Variables? I find the question hard to understand. – Yemon Choi Jun 1 '11 at 9:07
I don't really see why you need to use the letter $\lambda$ at all. It looks like you want a sequence of functions $a_n$ that depend only on $\rho$. – S. Carnahan Jun 1 '11 at 9:09
have you tried looking for finite sequences (with only 3, 4 or 5 terms) with $a_n(\lambda)$ given explicitely as, say, polynomial functions? – Olivier Bégassat Jun 1 '11 at 9:20
OK, so is $a_n(\lambda) = f(\lambda)b_n$ allowed, or not? If not, why not? Put more conditions on this $\lambda$ dependence! To be honest, this question looks like you're just making up random equations for its own sake; this can be great fun, but I don't see its relevance to mathematical research (unless you have a specific problem which needs this kind of construction, and even then it seems a bit too localised to me). – Zen Harper Jun 1 '11 at 10:29
I don't think this is a real question, since there is a large number of valid answers, which the OP does not like for some internal reason. Voting to close. – Igor Rivin Jun 1 '11 at 15:18

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 $\rho \gt 0.$
FWIW, I think the OP wants $\rho$ to be specified and the identity to hold for all $\lambda$ (though given how ill-defined the original question is, I am largely guessing. – Yemon Choi Jun 1 '11 at 19:04