series converges by comparison test

Question

I am currently studying series and using the comparison test to determine whether or not a series converges.

Is this an acceptable proof using the comparison test? (I have generalised a and b)

If we know that $a<b$ ($a$ and $b$ are functions of $i$)

$$\sum_{i=1}^\infty \frac{a}{b} < \sum_{i=1}^\infty\frac{b}{b} = 1 $$

Now since a series larger than a/b converges, using the comparison test we can say that the sequence a/b converges?

Thanks in regards.

Answer 1

$\sum_{i=1}^\infty\frac bb=\sum_{i=1}^\infty1$ obviously does not converge, so the comparison test cannot be used in this way.

Answer 2

Not only is your use of the comparison test incorrect as pointed out in the comments and answers, there is actually no way to determine if $\sum_{i=1}^\infty \frac{a_i}{b_i}$ exists.

Take $a_i = 1$ and $b_i=i^\alpha$. The series will converge if and only if $\alpha>1$.