A question on Comparison Test. Comparison Test states:

Suppose that we have two series  $\sum a_n$ and $\sum b_n $  with $a_n, b_n \geq 0$ for all $n$ and  $a_n \leq b_n$ for all $n$.  Then,
If  $\sum b_n$ is convergent then so is $\sum a_n$.
If  $\sum a_n$ is divergent then so is  $\sum b_n$.

I am just wondering, must $a_n \leq b_n$ be true for every $n$? Wouldn't it be possible if $b_n \leq a_n$ for finitely many terms, but the rest we still get convergence?
So if say the first $1000$ terms, we have $a_n > b_n$, but if there was a large enough $N$ giving us $a_n \leq b_n$ for $n \geq N$, wouldn't we still get the comparison test
$$\sum_{n \geq N} a_n \leq \sum_{n \geq N} b_n$$
to work?
added example
Here is a classic example.
If $a_n \geq 0$, $\sum a_n$ converges, so does $\sum a_n^2.$
Note for $n \geq n_0 \implies a_n < 1 \implies a_n^2 < a_n.$
So $$\sum_{n = n_0}^{\infty} a_n^2 < \sum_{n = n_0}^{\infty} a_n$$ converges because we don't care about what happens for $n < n_0$
 A: You are right: The reason we can ignore finitely many terms is because of the way we define the converge of a series. When we say $\sum_{n=1}^\infty b_n$ converges, we mean the sequence of partial sums, where $B_n = \sum_{i=1}^n a_i$, converges.
Remember the Cauchy Criteria for convergence. The sequence $B_n$ converges if for all $\epsilon > 0$, there exists an $N$ such that for all $m,n \geq N$, $|B_n - B_m| < \epsilon$. 
Thus if $m > n$, we have that $|B_n - B_m| = |b_{n+1} + ... + b_m| < \epsilon$. Now, lets look at a sequence $a_n$. As long as $a_n \leq b_n$ for $n \geq N$, the series $\sum a_n$ (or the sequence of partial sums $A_n$) is still able to meet the Cauchy Criterion: $$|A_n - A_m| = |a_{n+1} + ... + a_m| \leq |b_{n+1} + ... + b_m| < \epsilon$$
Do you see why the comparison test only works if $a_n, b_n \geq 0$?
A: You are correct you can change finitely many terms and the convergence test is unaffected. And probably one of the most general statements is that if you consider the partial sums $A_n$, $B_n$, then all you need is that $A_n \leq c B_n$ for infinitely many $n$ for some constant $c > 0$. So for example you could replace every pair of consecutive terms with two copies of the maximum of the two terms, and still get convergence. Of course for this to work you need all non-negative terms in both series.
