How do you do this question?

Answers:

Based on the limit comparison test, the series is convergent

========================================================

Explanation:

As n gets really large, the terms and are effectively the same. Taking the limit to infinity of the ratio leads to 1.

In other words,

That -2 doesn't really play a role when n gets very large.

This result is then multiplied by the 6 in the sequence, getting 6 as the final limit we want.

So,

Since the limiting value is positive and not infinity, this means that both series and converge or they both diverge.

----------------------------

If we plug n = 2 into the sequence, we get

The first term is a = 1/9

If we plugged n = 3, we would get

This shows that the common ratio is r = 1/3

Because -1 < r < 1 is true, we know that the infinite geometric series converges. Therefore, converges. We don't need to find the converging value.

Going back to the limit comparison test, we stated that both and either diverge together or converge together.

We've shown that the b series converges, so the 'a' series must converge as well.