Sequences where $\sum\limits_{n=k}^{\infty}{a_n}=\sum\limits_{n=k}^{\infty}{a_n^2}$

Question

I was recently looking at the series $\sum_{n=1}^{\infty}{\sin{n}\over{n}}$, for which the value quite cleanly comes out to be ${1\over2}(\pi-1)$, which is a rather cool closed form.

I then wondered what would happen to the value of the series if all the terms in the series were squared.

Turns out... nothing happens!

$\displaystyle\sum_{n=1}^{\infty}{\left({\sin{n}\over{n}}\right)}^2=\sum_{n=1}^{\infty}{\sin{n}\over{n}}={1\over2}(\pi-1)$.

This is a rather cool result, and I was wondering if there are any other simple series that share this property? Or, more generalized, series for which raising the terms to the power $m$ yields the same result as raising them to the power $p$.

Answer 1

This a long comment not an answer.

Notice that a similar property holds for the related integral

$$ \int_{0}^{\infty}\frac{\sin x}{x}dx = \int_{0}^{\infty}\bigg(\frac{\sin x}{x}\bigg)^2 dx = \frac{\pi}{2}. $$

This makes me wonder if such sequence has to do something with the property of orthogonality. Two functions $f$ and $g$ are said to be orthogonal with weight $1$ $$ \int_{0}^{\infty}f(x)g(x) = 0 $$

If we impose the condition that $g(x) = 1 - f(x)$ then we are looking at a special case of orthogonality where $$ \int_{0}^{\infty}f(x)(1-f(x)) = 0 $$

which is analogous to OP question since $\sum_{n=k}^{\infty}{a_n}(1-a_n) = 0$

Answer 2

This is one way of looking at these sequences. As your starting $k$ might not be $0$, we're inside a linear subspace of $l^2$, the space of sequences $(a_0, a_1, \ldots)$ whose sum $\sum_{n=0}^{\infty} |a_n|^2$ converges. As we also need the sum $\sum_{n=0}^{\infty} a_n$ to converge, that places us inside a somewhat smaller linear subspace $V$ of $l^2$.

In there, we're looking for sequences $a = (a_0, a_1, \ldots)$ such that $f(a) = 0$, where $f(a) = \sum_{n=k}^{\infty} a_n - \sum_{n=k}^{\infty} a_n^2$. This is a smooth function on $V$, and its differential at $a$ in the direction of $b$ is $$ d_b f(a) = \sum_{n=k}^{\infty} b_n - 2 \sum_{n=k}^{\infty} a_n b_n. $$ If $a$ is the zero sequence in $V$ (that is, any sequence $a$ with $a_n = 0$ for $n \geq k$), then $df(a) = 0$. If $a$ is not the zero sequence, then there exists a $b$ such that $d_bf(a) \not= 0$, take for example $b = -\frac12 a$.

Thus $f$ is a smooth function whose differential is nondegenerate away from the linear subspace $N$ of $V$ defined by the zero sequences in $V$, and degenerate on that subspace. This means that the set $X = \{a \in V \mid f(a) = 0 \}$ of sequences that satisfy your identity is an infinite-dimensional hypersurface that is nonsingular outside of $N$, and has singularities on that set.

Gerry's comment shows that there is also a smooth projection function $p : V \setminus N \to X \setminus N$, that is, a smooth function $p$ such that $p(a) = a$ for any sequence $a \in X$.