Convergence test for partial sum whose elements all change as the index increases

Question

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.

I just want to make sure I understand. I'm think of this as an infinite lower triangular matrix, where $S_n$ is the sum of the $n$th row. When you say that each element of the sequence goes to $0,$ it means that each of the columns goes to $0$. Then you want to know whether that guarantees that the row sums go to a finite limit. Am I correct? — saulspatz, Aug 02 '18 at 02:17
This cannot be answered in general without more specifics or, even better, some example(s). each element of the sequence is a function also of the length of the sequence This is always the case for Riemann sums for example, see e.g. How do you calculate this limit $\lim_{n\to\infty}\sum_{k=1}^{n} \frac{k}{n^2+k^2}$?, but this is by no means a general answer. — dxiv, Aug 02 '18 at 02:42

score 1 · Accepted Answer · answered Aug 02 '18 at 02:27

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 $\log n.$

Convergence test for partial sum whose elements all change as the index increases

1 Answers1