Sum of a series

In summary, the conversation discusses a series with coefficients satisfying a recurrence relation and the concept of an "optimum number of terms" for convergence. The question asks how to obtain this number and the sum of the series, with a suggestion of using a brute force algorithm. However, without a clear definition of optimum, it is impossible to answer the question.
  • #1
lokofer
106
0
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:
 
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  • #2
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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  • #3
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.
 

What is the "Sum of a Series"?

The "Sum of a Series" is a mathematical concept that refers to the total value obtained by adding all the terms in a specific sequence or pattern.

How is the sum of a series calculated?

The sum of a series is calculated by adding each term in the series together, starting from the first term to the last term. This can be done using a formula or by manually adding the terms.

What is the difference between a finite and infinite series?

A finite series has a limited number of terms, while an infinite series has an infinite number of terms. This means that the sum of a finite series will eventually reach a final value, while the sum of an infinite series will continue to increase without ever reaching a final value.

What is the significance of the sum of a series in real-world applications?

The sum of a series has various applications in fields such as physics, engineering, and finance. For example, it can be used to calculate the total distance traveled by an object with changing velocity, or to determine the total value of an investment with compound interest.

How can the sum of a series be used to find missing values?

The sum of a series can be used to find missing values by rearranging the formula and solving for the unknown term. This is known as "solving a series". It can also be used to check the accuracy of a given value in a series by comparing it to the calculated sum.

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