Open-form and Closed form

Question

How can an open-form of recurrence be converted into its equivalent closedform. Furthermore, what are some commonly used closed-forms that are usually used efficiently.

Answer 1

any mathematical formula that is in series or in sequence form, how can we converted this series or sequence in closed form

Answer 2

I think you are talking about recursive functions and math.

e.g. consider the following sum recursive function

sum(0) = 0
sum(i) = sum(i-1) + i

this form is not closed. A closed form is sum(n) = (n+1)*n/2, where you only use basic operations like +-*/, power, and sometimes factorial.

For your question, how to convert a open-form formula into a closed form. The answer is there is no general rule to transform all open-form into closed form because some of the open forms don't have equivalent closed forms.

You may refer to Concrete Mathematics for a serious treatment of this subject. The book's primary goal is to convert a large family of recursive function/open forms into their closed forms.