I have a sequence of length $n$, $\{b_i(n),\, i=1,…,n\}$. Each $b_i(n)$ is a strictly positive rational number. I write $b_i(n)$ because indeed each element of the sequence is a function also of the length of the sequence, and as the sequence progresses all its elements change value.
If $S_n = \sum_{i=1}^nb_i(n)$, I need to examine whether $\lim_{n\to \infty}S_n$ converges to a finite limit or not. Partial sums look like
$$S_n = \sum_{i=1}^nb_i(n)$$
$$S_{n+1} = \sum_{i=1}^nb_i(n+1) + b_{n+1}(n+1)$$
I know that the limit of the partial sum will never be zero. I know that each element of the sequence tends to zero as $n$ progresses. But since I have infinite elements of the partial sum that go to zero, I am not sure how to determine whether the whole infinite series remains finite or not.
Can somebody suggest any method or point to relevant sources that examine this issue? I am not sure that the usual tests for convergence of a series/limit of a partial sum (like the ratio test etc) apply here.
Best Answer
If I've understood the question correctly, as indicated in my comment, then the answer is "no." For example set $$ b_i(n)=\cases{{1\over n},&$1\leq i\leq n,\ $if $n$ is odd\\ {2\over n},&$1\leq i\leq n,\ $if $n$ is even} $$ so that $S_n=1$ if $n$ is odd, and $S_n=2$ is $n$ is even, and $S_n$ does not converge.
Or you could define $$b_i(n) ={\log n\over n}$$ to get $S_n\to\infty.$
EDIT
I just remembered that you want $b_i(n)$ to be rational. That makes no difference. Just take a good rational approximation to $\log n.$