How Can We Determine the Optimum Number of Terms for a Convergent Series?

[lokofer]

Let be the series...$$S=\sum_{n=0}^{\infty} a(n)$$

where a(0)=1=a(1) and the rest of coefficients satisfy a recurrence relation (linear or non-linear) so $$F(n,a_{n+2} , a_{n+1},a_{n})=n$$ ..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...

 

[matt grime]

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.

 

[CRGreathouse]

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.