Sequences where ∑n=k∞an=∑n=k∞an2 I was recently looking at the series ∑n=1∞=sin⁡nn, for which the value quite cleanly comes out to be 12π−1, which is a rather cool closed form.

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Sequences where ∑n=k∞an=∑n=k∞an2  I was recently looking at the series ∑n=1∞=sin⁡nn, for which the value quite cleanly comes out to be 12π−1, which is a rather cool closed form.

trnovitom06 · Accepted Answer

Notice that a similar property holds for the related integral  ∫0∞sin⁡xxdx=∫0∞(sin⁡xx)2dx=π2  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  ∫0∞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  ∫0∞f(x)(1−f(x))=0  which is analogous to question since ∑n=k∞an(1−an)=0

bemolizisqt · Accepted Answer

This is one way of looking at these sequences. As your starting k might not be 0, we&#039;re inside a linear subspace of l2, the space of sequences (a0,a1,…) whose sum ∑{n=0}∞|an|2 converges. As we also need the sum ∑{n=0}∞an to converge, that places us inside a somewhat smaller linear subspace V of l2 In there, we&#039;re looking for sequences a=(a0,a1,…) such that f(a)=0, where f(a)=∑{n=k}∞an−∑{n=k}∞an2. This is a smooth function on V, and its differential at a in the direction of b is xdbf(a)=∑n=k∞bn−2∑n=k∞anbn If a is the zero sequence in V (that is, any sequence a with an=0 for n≥k), then df(a)=0. If a is not the zero sequence, then there exists a b such that dbf(a)≠0, take for example b=−12a. 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∈V∣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. there is also a smooth projection function p:VN→XN that is, a smooth function p such that p(a)=a for any sequence a∈X.

Sequences where \sum_{n=k}^{\infty} a_n=\sum_{n=k}^{\infty}a^2_n I was recently looking at the series

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