For questions about recurrence relations, convergence tests, and identifying sequences

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5
votes
0answers
55 views
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Show the equivalence of two infinite series over Bessel functions

The following sums pop up in diffraction theory and are related to Lommel's function of two variables. Let $u,v\in\mathbb{R}$. I claim that $$\sum_{n=0}^\infty i^n \left ( \frac{u}{v} \right )^n ...
9
votes
6answers
261 views
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When are two series the same?

A series is an expression of the form $$ \sum_{n=k}^{\infty} a_n $$ where the $a_n$ are real numbers and they depend on $n$. If $a_n = b_n$ for all $n\geq k$, then I would assume that one would say ...
1
vote
0answers
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An upper bound for $\sum_{n,m} \left(|a_n\phi_n|^2+|a_m\phi_m|^2\right) \left|K_{n-m}\right|$.

Let us consider $a_n, \phi_n, K_{n}$ complex sequences. Let $$\sum_{n,m} \left(|a_n\phi_n|^2+|a_m\phi_m|^2\right) \left|K_{n-m}\right|$$ where $\left|K_{n}\right|\leq \gamma$ and $\gamma>0$. Can we ...
0
votes
0answers
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Find the finite sequence that minimizes the value of $T_5(P)$

Given a finite sequence $P(a_1,b_1),(a_2,b_2),...,(a_n,b_n)$, define $T_1(P):=a_1+b_1$, $\forall 2\leq k\leq n$, $T_k(P)=b_k+\max\{T_{k-1}(P),a_1+a_2+...+a_k\}$. Let $m=\min\{a,b,c,d\}$. ...