Relation between sequence and sums

Question

I have two questions regarding sequences and sums.

$1)$Given a recurrence sequence, does it exist, and can be found, a formula for the nth term?
$2)$Does there exist a bijection between the general term of a sum and the general term of a sequence?

Answer 1

1) Not always. Solving recurrence relations is, in general, hard or impossible.

2) Yes, if I understand you correctly. Consider the sequence $(a_n)_n$ where $a_n = \sum_{i=1}^n b_i$ to show that every series induces a sequence; on the other hand, consider the sum whose partial sums are $b_n = \sum_{i=1}^n \left[a_i - b_{i-1}\right]$ to show that every sequence $(a_n)_n$ can be written as the partial sums of a series.

Answer 2

There are many such recurrence sequences of which the general term is unknown. For example take $a_{n+1}=(1-a_n)^{a_n}\sin a_n$. For your second question the answer is affirmative. Since all the terms of a sequence participate in the general term of summation equally likely there exists a bijection whose the equation has the same definition of summation