Vrftjkry

I have got a sequence $frac19,frac{7}{17},frac{17}{25}, cdots$ and got two different version for general term:

$$frac{2n^2-1}{8n+1} & frac{8(n-1)+(-1)^{n+1}}{8n+1}$$

but uisng the 1st one we get the sequence is divergent and the second one gives us convergence!!

I am really confused!

These two expressions only agree until $n=3$. At $n=4$, the first expression gives $frac{31}{33}$, whereas the second gives $frac{23}{33}$. As $n to infty$, these expressions are giving you two different sequences. So one will converge and the other won't but their behaviors don't have to be dependent on each other just because the first few terms are a match.

This is possible, finitely many terms do not tell us about the convergence of the whole sequence.

Also, in general, we can't tell the whole sequence from finitely many terms.

You have made up your formulas based on just three terms. Others may come up with other formulas or just make up other terms for the rest of the sequence.

You only have three terms of your sequence, depending on what happens with other terms everything is possible.

For example you can define the rest of your sequence as $$4,5,6,...n,...$$ to make it divergent or $$ 0,0,0,...$$ or anything else.