Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I have a recurrent equation that defines a sequence $a_k$ from a sequence $b_k$:

$$a_k = b_k - \sum^\infty_{i=k+1}a_i$$

How can I write this equation without mentioning $a_k$ on the right side?

share|improve this question
    
The given equation easily defines $b_k$ in terms of a summation over terms $a_i$ from $i=k$ to infinity. Are you trying to "solve" this relationship to obtain $a_k$ in terms of $b_i$'s? –  hardmath Jun 8 at 12:32
    
@hardmath Yes, that's what I'm trying to do. I just saw myself how trivial this question is. –  FUZxxl Jun 8 at 12:34
    
@FUZxxl: you'd better rephrase your question: there is no $a_k$ on the RHS ! (Just a few $a_i$'s, but no $a_k$ of any kind). –  Yves Daoust Jun 8 at 12:36
    
Obviously, the $b_k$ are the "integral" of the $a_k$, so that the $a_k$ are the "derivative" of the $b_k$. –  Yves Daoust Jun 8 at 12:40

1 Answer 1

up vote 4 down vote accepted

$$b_k = a_k + \sum_{i=k+1}a_i = \sum_{i=k}a_i$$

$$b_{k-1} - b_k = \sum_{i=k-1}a_i - \sum_{i=k}a_i = a_{k-1}$$

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.