Sum of a series (1 Viewer)

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Let be the series....[tex] S=\sum_{n=0}^{\infty} a(n) [/tex]

where a(0)=1=a(1) and the rest of coefficients satisfy a recurrence relation (linear or non-linear) so [tex] F(n,a_{n+2} , a_{n+1},a_{n})=n [/tex] :tongue2: :tongue2: ..then my question is let's suppose that the series has an "optimum number of terms" K so if you take k-terms the series converges to a optimum value, otherwise the series (taking all terms) diverges) my question is how would we obtain this k and the sum of the series... a "brute force" algorithm would say that you take a big number of terms and solve the recurrence by using a computer...:rolleyes: :rolleyes:
 

matt grime

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Since you have not defined optimum the question is impossible to answer.

When you use K and k are we supposed to think they refer to the same thing? What is a k-term? Do you just mean sum the first k terms (so why use the word convergent for a finite sum?), or do you mean to pick some infinite subset of the terms?

Your recurrence relation could very well be easy to solve (it is only a second order recurrence relation, as written.
 
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CRGreathouse

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The sum of a finite number of finite terms always converges. If your ininite sum is divergent, then any stopping point will put you in the situation you mention.
 

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