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

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3
votes
1answer
245 views
+50

Prove or disprove that there exists a unique positive integer sequence $\{a_{n}\}$ satisfying a condition

Question: Prove or disprove: there exists a unique positive integer sequence $\{a_{n}\}$ satisfying the following condition: $\forall m\in N^{+}$, there exists a unique integer sequence ...
7
votes
2answers
341 views
+100

Is there an easier way to find the “natural” integration constant?

Suppose we take consequtive derivatives of a function at a point and then interpolate them with Newton series (Newton interpolation formula) so to obtain a smooth curve. ...
4
votes
1answer
139 views
+50

How prove this sequence $u_{m}=v_{m}$

Question: Assume that $m$ is a positive integer, define the sequence $$\{u_{k}\},\{v_{k}\},u_{0}=v_{0}=u_{1}=v_{1}=1$$ and for any real number $a_{i},i=\{1,2,\cdots,m-1\}$, $$\begin{cases} ...
3
votes
0answers
44 views
+100

'Deriving' the Laplace Transform from the $z$ Transform: Missing a $\Delta t$

Textbooks normally give the following 'derivation' (or justification, if you prefer) of the z-Transform from the Laplace Transform. Let $x(t)$ be a signal defined on $t\geq 0$, and write a discretized ...