sequence of series approximating another series
Solution 1:
Take $c=\sum_{i=0}^{+\infty}0$ and $a_n=\sum_{i=0}^{n-1}0+\sum_{i=n}^{+\infty}\frac{1}{2^{i-n}}$. Then $c$ and $a_n$ have the same first $n$ terms but $$\lim_{n\to+\infty}c-a_n=0-2=-2\neq 0.$$